critical ideals of graphs - cinvestav · g], the determinantal ideals of l(g;x g) are ideals on z[x...

Center ofResearch andAdvanced Studies

of the National Polytechnic Institute

Campus Zacatenco

Department of Mathematics

Critical ideals of graphs

A Dissertation presented by

Hector Hugo Corrales Sanchez

to obtain the degree of

Doctor in Science

in the speciality of Mathematics

Thesis Advisor: Ph.D. Carlos E. Valencia O.

Mexico, D.F. September 2014

Agradecimientos

Agradezco al CONACyT el apoyo recibido durante la elaboracion de este trabajo,

primeramente como becario y despues bajo el proyecto SEP-CONACyT 166059.

Abstract

The Laplacian matrix of a graph G is defined as L(G) = D(G) − A(G) where A(G) is the adjacency

matrix of G and D(G) his degree matrix. The critical group of G, denoted K(G), is the torsion part

of the cokernel of L(G). We begin this thesis generalizing this definition: if M is a m × n matrix

with entries over a commutative ring A we define his critical module, which we denote K(M), as

the torsion part of Am/MAn. In particular, we study K(M) for M ∈ Mn(Km(Z)) where Km(Z) =

{(a+ b)In + bA(Km)|a, b ∈ Z}. This allow us to determine the structure of the critical group of some

variations of bipartite graphs.

Motivated by our results we introduce the critical ideals of a graph. The intention is find common

properties of critical groups associated to graphs with a common structure. We define the generalized

Laplacian matrix of a graph G with n vertices as the matrix L(G,XG) = D(XG)−A(G) where XG is

the set of n undetermineds indexed by the vertices of G. Since L(G,XG) is a matrix with entries over

Z[XG], the determinantal ideals of L(G,XG) are ideals on Z[XG] which we call critical ideals of G.

Next we study how the critical ideals encode the combinatorial information of a graph. We present

explicit the critical ideals of the paths, the cycles and the complete graph and show that in each case

the given set of generators is a Groebner basis under degree lexicographic order. Also, we investigate

how the structure of a graph can be used to determine a generating sets for the critical ideals. In this

sense, we show that if the graph is a tree then, the set of 2-matchings can be used to give generators

for the critical ideals. In particular, if ν2(T ) is the maximum size of a 2-matching on a tree T , then T

exactly ν2(T ) critical ideals are trivial.

Introduction

Let G = (V,E) be a finite graph, m(u,v) the number of edges from u to v, and A(G) be the adjacency

matrix of G given by A(G)u,v = m(u,v). The Laplacian matrix of G is given by L(G) = D(G)−A(G),

where D(G) is the diagonal matrix with the degrees of the vertices of G in the diagonal entries. The

critical group of G, denoted K(G), is the torsion part of the cokernel of L(G).

The critical group is an abelian group isomorphic to the sandpile group defined independently by

Dhar [23], motivated by the abelian sandpile model introduced by in Bak [7], and by Lorenzini [31] in

connection with arithmetic geometry. In different contexts this group is also named Jacobian group or

group of components.

For the past thirty years several authors studied critical groups from different perspectives but, among

the published works theres been a lack of general results about the behavior of critical groups over

general graphs. The most general result were obtained by Reiner et. al. [15, 26, 9]. Almost all the

remain work published about critical groups has the same approach: define a family of graphs and

describe the critical group associated to each graph. The result is a family of groups with a similar

structure, see for instance [6, 11, 13, 14, 25, 28, 33, 34, 37, 38, 39]. This works has one more thing in

common, they use the Smith normal form to obtain the invariant factors of K(G).

We begin this thesis with a similar approach. First, we generalize the concept of critical group to any

matrix with entries on an arbitrary commutative ring with identity A. More precisely, for M ∈Mm×n(A)

his critical module, denoted K(M), is defined as K(M) = An/MAm. After skipping the technical issues

of this generalization we put special attention to matrices that generalize the Laplacians of the path, the

cycle and the complete graph. Among this type of matrices the ones associated to complete graphs have

an interesting property: they form a subalgebra of Mn(A) which we denote Kn(A). As Theorem I.2.5

show, if M ∈Mn(Km(A)) then, K(M) ∼= K(A1)m−2 ⊕K(A2) for some A1∈Mn(A) and A2∈M2n(A).

Using this result we derive the structure of critical groups associated to several graphs.

Although this approach is useful in many specific cases, it seems few feasible to obtain results beyond

very symmetric graphs. Once one look carefully it seems obvious the reason: the graph G is used only

viii

to construct the Laplacian matrix. So, we look the problem almost exclusively as an algebraic problem.

It seems that one forgets the crucial role of algebraic combinatorics in present mathematics. This was

the main reason which led us to define critical ideals. Roughly speaking the purpose of critical ideals is

separate the steps involved in the Smith normal form of a matrix, specially the arithmetic involved on

it. As Chapter I shows, one can exploit the structure of Laplacian matrices looking the combinatorial

properties inherited from the graph.

Keeping in mind this objective in Chapter II we introduce the generalized Laplacian of a graph G as

L(G,XG) = D(XG) − A(G) where XG = {xv|v ∈ V (G)} is a set of undetermineds and D(XG) is the

diagonal matrix with XG in the main diagonal. In view of the characterization of the invariant factors

of K(G) given by the Smith normal form we are interested in the determinaltal ideals of L(G,XG).

This are precisely the critical ideals of G. If |G| = n then, for each 1 ≤ i ≤ n there is a i-critical ideal

corresponding to the i-minors of L(G,XG). We begin Chapter II presenting the general behavior of

critical ideals and their relation with critical groups. An elementary but remarkable fact is that the

number of trivial critical ideals, which we denote γ(G), is related to the clique number ω(G) and the

stability number α(G) by the relation γ(G) ≤ min{2(n − α(G)), 2(n − ω(G)) + 1}. We also describe

the critical ideals of the paths, the complete graphs and the cycles with the purpose of illustrate how

the combinatorics of G reflects on their critical ideals.

Chapter III looks deeper in the relation between critical ideals and the structure of the graph. This

chapter is devoted to present an interesting connection that occurs when the graph is a tree T . We will

show how the set of 2-matchings on T ` (which denote T with a loop added on each vertex) describe

the critical ideals of T , and even more, a special kind of 2-matchings (which we call minimal) gives

a minimal set of generators for each critical ideal. As result, γ(T ) equals the size of a maximum 2-

matching on T . Another consequence is that if T has n vertices it turns that the minimal 2-matchings

of T ` with size n − 1 form a Groebner basis for the n − 1 critical ideal. Surprisingly the reduction

process involved relays only on the structure of T .

We finalize this thesis showing some applications of the obtained results in Chapter III. First we describe

the critical ideals of trees with depth one and two, next we derive information about the critical group

of wired regular trees and finally we present a connection with arithmetic geometry using arithmetical

graphs.

Contents

Agradecimientos iii

Abstract v

Introduction vii

Preliminaries 1

1 Smith Normal Form 1

2 Critical Group of a graph 2

3 Grobner Basis 2

Chapter I : Critical group of matrices 5

1 The critical module of matrices 6

2 Some applications 11

The subring Kn(A) 13

Chapter II : Critical ideals of graphs 25

1 Critical ideals of graphs 26

The invariant γ 29

Critical ideals of the complete graphs 32

Critical ideals and the characteristic polynomials 34

2 Critical ideals of the cycle 35

Critical ideals of the cycle 39

x

Chapter III : Critical ideals of trees 49

1 2-matchings of trees 50

Two matchings of G` 53

2 Critical Ideals of Trees 54

The non-vanishing minors of L(T,XT ). 55

3 Groebner basis of critical ideals 59

4 Applications on critical group 63

Trees of depth one and two 63

Wired d-regular trees 65

Arithmetical trees 68

Bibliography 71

Index 75

Preliminaries

The aim of this section is to provide a self contained introduction for the concepts that will be used

with frequency on the rest of this thesis. We adopt standard notation and names of graph theory,

group theory and commutative algebra.

In the next, a graphs means a finite connected graph, a multigraph will be a finite connected multigraph

and similar for digraphs and multidigraphs.

1. Smith Normal Form

Given a matrix M ∈Mm(Z), the cokernel of M , denoted coker(M), is defined as coker(M)=Zm/MZm.

Since Z is a Bezout Domain, M is equivalent to a unique matrix D = diag(d1, . . . , dk, 0, . . . , 0) with

d1, . . . , dk ∈ Z>0 and d1| · · · |dk. Let U, V ∈ GLn(Z) such that UMV = D, since V is invertible

U(MZn) = DZn, and since U is invertible

coker(M) = Zm/MZm coker(D) = Zm/DZm T ⊕ Zn−k-∼= -

∼=,

where T = Zk/diag(d1, . . . , dk)Zk is a finite group. T and Zn−r are called the torsion and the free parts

of coker(M) respectively. The unique diagonal matrix D is called the Smith Normal Form of M .

Let I = {i1, . . . , ir} ⊆ [n], and J = {j1, . . . , js} ⊆ [n]. The submatrix of M formed by rows i1, . . . , ir

and columns j1, . . . , js is denoted by M [I; J ]. On the other hand, the submatrix obtained from M by

deleting rows i1, . . . , ir and columns j1, . . . , js will be denoted by M(I; J). That is, M(I; J) = M [Ic; Jc].

If |I| = |J | = r, then M [I; J ] is called an r-square submatrix or a square submatrix of size r of M .

An r-minor is the determinant of an r-square submatrix. The set of i-minors of a matrix M will be

denoted by minorsi(M).

The Smith Normal Form diag(d1, . . . , dk, 0, . . . , 0) of M is characterized by

k = max{i |minorsi(M) 6= {0}

},

di = gcd(minorsi(M)) for each 1 ≤ i ≤ k.

2 Preliminaries

2. Critical Group of a graph

For a multigraph G with n vertices his adjacency matrix is defined as the square matrix of size n given

on each pair of vertices u and v by

A(G)u,v = −muv = − (number of multiple edges between u and v),

and the Laplacian matrix, denoted L(G) is defined by L(G) = D(G) − A(G) where D(G) is a diago-

nalmatrix given by D(G)v,v = dG(v).

The critical group of G, denoted K(G), is defined as the torsion part of coker(L(G)).

Kirchoff matrix-tree theorem [12, Theorem 2.12] ensures that minorsn−1(L(G)) 6= {0}. Furthermore, if

L(G, s) is a principal submatrix of L(G) that result by removing the column and row corresponding

to a vertex s of G, then coker(L(G)) ∼= coker(L(G, s)). L(G, s) is called a reduced Laplacian matrix

Since L(G) has rank n− 1 the Smith normal form of L(G) has the form diag(f1, . . . , fn−1, 0) and

K(G) ∼= Zf1 ⊕ · · · ⊕ Zfn−1 .

The integers f1, . . . , fn−1 are called invariant factors of K(G). Since Z1 is the trivial group, if fk = 1

for some k = 1, . . . , n− 1 then, we say that K(G) has at least k invariant factors and if fk+1 6= 1 then,

we say that K(G) has exactly k invariant factors.

3. Grobner Basis

Usually the theory of Grobner basis deals with ideals in a polynomial ring over a field. However, in

this thesis we deal with ideals in a polynomial ring over the integers. There exists a theory of Grobner

basis over almost any kind of rings.

We recall some basic concepts on Grobner basis, for more details see [1]. First, let P be a principal

ideal domain. A monomial order or order term in the polynomial ring R = P[x1, . . . , xn] is a total

order ≺ in the set of monomials of R such that

(i) 1 ≺ xα for all 0 6= α ∈ Nn, and

(ii) if xα ≺ xβ, then xα+γ ≺ xβ+γ for all γ ∈ Nn,

where xα = xα11 · · ·xαnn .

Now, given an order term ≺ and p ∈ P[X], let lt(p), lp(p), and lc(p) be the leading term, the leading

power, and the leading coefficient of p, respectively. Given a subset S of P[X] its leading term ideal of

S is the ideal

Lt(S) = 〈lt(s) | s ∈ S〉.

A finite set of nonzero polynomials B = {b1, . . . , bs} of an ideal I is called a Grobner basis of I with

respect to an order term ≺ if Lt(B) = Lt(I). Moreover, it is called reduced if lc(bi) = 1 for all 1 ≤ i ≤ sand no nonzero term in bi is divisible by any lp(bj) for all 1 ≤ i 6= j ≤ s.A good characterization of Grobner basis is given in terms of the so called S-polynomials.

Preliminaries 3

Definition. Let f , f ′ be polynomials in P[X] and B be a set of polynomials in P[X]. We say that f

reduces strongly to f ′ modulo B if

• lt(f ′) ≺ lt(f), and

• there exist b ∈ B and h ∈ P[X] such that f ′ = f − hb.

Moreover, if f∗ ∈ P[X] can be obtained from f in a finite number of reductions, we write f →B f∗.

That is, if f =∑t

j=1 pijbij + f∗ with pij ∈ P[X] and lt(pijbij ) 6= lt(pikbik) for all j 6= k, then f →B f∗.

Now, given f and g polynomials in P[X], their S-polynomial, denoted by S(f, g), is given by

S(f, g) =c

cf

X

Xff − c

cg

X

Xgg,

where Xf = lt(f), cf = lc(f), Xg = lt(g), cg = lc(g), X = lcm(Xf , Xg), and c = lcm(cf , cg).

The next Lemma, knowm as Buchberger’s criterion, gives us a useful criterion for checking whether a

set of generators of an ideal is a Grobner basis.

Lemma. Let I be an ideal of polynomials over a PID and B be a generating set of I. Then B is a

Grobner basis for I if and only if S(f, g)→B 0 for all f 6= g ∈ B.

In this tesis we only work with the so called degree lexicographic order.

Definition 3.1. Let P[x1, . . . , xn], α, β ∈ Nn; then xα ≺ xβ if

• α1 + · · ·+ αn < β1 + · · ·+ βn,

• or α1 + · · ·+ αn = β1 + · · ·+ βn and exist i = 1, . . . , n such that

α1 = β1, α2 = β2, . . . , αi−1 = βi−1 and αi < βi.

Chapter I : Critical group of matrices

The concept of critical group can be generalized easily to an arbitrary commutative ring with identity

A. More precisely, if M ∈Mm×n(A), then the critical module of M , denoted by K(M), is defined as:

K(M) := An/MAm.

Given H < GLn(A) and H ′ < GLm(A), we say that M,N ∈Mm×n(A) are (H,H ′)-equivalent, denoted

by N ∼(H,H′) M , if there exist P ∈ H and Q ∈ H ′ such that N = PMQ. When H = SLn(A)

and H ′ = SLm(A), then we simply say that M and N are unitary equivalent and will be denoted by

N ∼u M . Also, if H and H ′ are the subgroups generated by the elementary matrices, we simply say

that M and N are elementary equivalent and will be denoted by N ∼e M . Finally, if H = GLn(A)

and H = GLn(A), then we simply say that M and N are equivalent and will be denoted by N ∼A M

or M ∼ N if the ring A is clear from the context.

Is not difficult to see that if M and N are equivalent, then

K(M) = An/MAm ∼= An/NAm = K(N).

When the base ring A is a Principal Ideal Domain (PID), another description of the critical group of

a matrix M is given by

K(M) =

|V |⊕i=1

A∆i(M)/∆i−1(M),

where ∆i(M) is the greatest common divisor of all the i-minors of M .

This chapter is divided in two sections: In the first section, we find diagonal matrices over an arbitrary

commutative ring with identity that are equivalent to some matrices that generalize the Laplacian

matrices of the path, the cycle, and the complete graph. In the second section we apply the results

obtained in the first section in the case when the base ring is the ring of integers and some subrings of

matrices. In particular we are able to calculate the critical group of the m-cones of the l-duplications of

the path, the cycle, the complete graph, the bipartite complete graph with m vertices in each partition,

the bipartite complete graph with 2m vertices minus a matching, etc.

6 Chapter I : Critical group of matrices

1. The critical module of matrices

In this section we will find diagonal matrices that are equivalent to some matrices that generalize the

Laplacian matrices of the path, the cycle, and the complete graphs. After that, we will apply these

results to calculate the critical group for several families of graphs.

For all n ≥ 2 and a, b ∈ A, let Kn(a, b) = (a + b)In + bA(Kn), Tn(a, b) = aIn + bA(Pn), Cn(a, b) =

aIn + bA(Cn), and

Pn(a, b) =

a+b b 0 . . . 0

b a. . .

. . ....

0. . .

. . .. . . 0

.... . .

. . . a b

0 · · · 0 b a+b

.

where In ∈Mn×n(A) is the identity matrix on order n.

Since the critical module of a matrix is invariant under equivalency classes, then in order to determine

the critical module of a matrix, it is enough to find an equivalent diagonal matrix.

Theorem 1.1. Let a, b ∈ A such that the equation ax + by = 1 has solution in A and fn(x, y) are

polynomials in A[x, y] that satisfy the recurrence relation

fn(x, y) = xfn−1(x, y)− y2fn−2(x, y)

with initial values f−1(x, y) = 0 and f0(x, y) = 1. Then

(i) Tn(a, b) ∼u diag(1, . . . , 1, fn(a, b)) for all n ≥ 2,

(ii) Pn(a, b) ∼u diag(1, . . . , 1, (a+ 2b)fn−1(a, b)) for all n ≥ 2,

(iii) Kn(a, b) ∼u diag(1, a, . . . , a, a(a+ nb)) for all n ≥ 2, and

(iv) Cn(a, b) ∼u In−2 ⊕ C for all n ≥ 4, where

C =

fq(a, b)

a 2b

2b a

if n− 2 = 2q,

(fq+1(a, b)− bfq(a, b)

) 1 0

0 a+ 2b

if n− 2 = 2q + 1.

Proof. (i) For all l ≥ 2 and 1 ≤ k ≤ l − 1, let

Zk,l(a, b) =

fk bfk−1 0l−2

b

0 Tl−1(a, b)

∈Ml(A), and Zn,1(a, b) = (fn),

where fk := fk(a, b) for all k ≥ −1 and 0 is the matrix with all the entries equal to 0. Note that

Z1,n(a, b) = Tn(a, b)

Now we will prove the following statement:

Section 1 : The critical module of matrices 7

Claim 1.2. For all l ≥ 2 and 1 ≤ k ≤ l − 1

Zk,l(a, b) ∼u I1 ⊕ Zk+1,l−1(a, b).

Proof. Let x1, y1 ∈ A be a solution of the equation ax+ by = 1. Moreover, for all k ≥ 1, let

xk = xk1 and yk =

k∑i=1

(k

i

)aibk−1−ixi1y

k−i1 − xk1(fk − ak)/b,

that is, xk and yk are a solution of the equation fkxk + byk = 1.

Since xk yk 0

−b fk 0

0 0 In−k−1

Zk,l(a, b) =

1 ∗ ∗ ∗0 afk − b2fk−1 bfk 0

0 b

0 0 Tn−k−1(a, b)

and det

(xk yk

−b fk

)= fkxk + byk = 1, then Zk,l(a, b) ∼u I1 ⊕ Zk+1,l−1(a, b).

Applying Claim 1.2, we get that Tn(a, b) = Z1,n(a, b) ∼u In−1 ⊕ Zn,1(a, b).

(ii) For all n ≥ 0, m ≥ 1, let

Zk,l(a, b) =

fk + bfk−1 b(fk−1 + bfk−2) 0 · · · 0

b a b · · · 0

0 b 0

0 0 Tl−3(a, b) 0...

... b

0 0 0 b a+ b

∈Ml(A).

Also, let x′k = xk and y′k = yk−fk−1xk, that is, xk and yk are a solution of the equation (fk+bfk−1)x′k+

by′k = 1. Since

x′k y′k 0

−b fk + bfk−1 0

0 0 Im

Zk,l(a, b) =

1 ∗ ∗ · · · ∗0 fk+1 + bfk b(fk + bfk−1) 0 0

0 b 0

0 0 Tl−4(a, b) 0...

... b

0 0 0 b a+ b

for all l ≥ 2 and det

(x′k y′k−b fk + bfk−1

)= (fk + bfk−1)xk + byk = 1, then Zk,l(a, b) ∼u I1 ⊕

Zk+1,l−1(a, b) for all l ≥ 2. Thus

Pn(a, b) = Z1,n(a, b) ∼u In ⊕

(fn−1 + bfn−2 b(fn−2 + bfn−3)

b a+ b

).


Therefore, Pn+2(a, b) ∼u diag(1, . . . , 1, (a+ 2b)fn−1(a, b)) because(x′n−1 y′n−1

−b fn−1 + bfn−2

)(fn−1 + bfn−2 b(fn−2 + bfn−3)

b a+ b

)=

(1 ∗0 (a+ 2b)fn−1

)

∼e

(1 0

0 (a+ 2b)fn−1(a, b)

).

(iii) We begin proving the following statement:

Claim 1.3. If n ≥ 2, then Kn(a, b) ∼e

(a b

na −a

)⊕ aIn−2.

Proof. It turns out because

PnKn(a, b)Qn =

b b · · · b b a+ b

a a a · · · a −(n− 1)a

0 a −a 0 · · · 0

0 0 a −a · · ·...

......

. . .. . .

. . . 0

0 0 · · · 0 a −a

Qn

=

(a b

−na a

)⊕ aIn−2,

where

Pn=

0 0 · · · · · · 0 1

1 1 · · · 1 1 −n+ 1

0 1 −1 0 · · · 0

0 0. . .

. . .. . .

......

.... . .

. . .. . . 0

0 0 · · · 0 1 −1

and Qn=

−n+ 1 1 −1 −2 · · · −n+ 2

1 0 1 1 · · · 1

1 0 0 1. . .

...

1 0 0 0 1...

......

.... . .

. . . 1

1 0 0 · · · 0 0

are elementary matrices.

On the other hand, Kn(a, b) ∼u diag(1, a, . . . , a, a(a+ nb)) because(a b

−na a

)(x1 −by1 a

)=

(1 0

∗ a(a+ nb)

)∼e

(1 0

0 a(a+ nb)

)

and det

(x1 −by1 a

)= ax1 + by1 = 1.

(iv) For all l ≥ 4 and k ≥ 1, let

Zk,l(a, b) :=

fk bfk−1 0l−4 b2fk−2 bfk−1

b 0

Tl−2(a, b)

0 b

bfk−1 b2fk−2 0l−4 bfk−1 fk

∈Ml(A)

Section 1 : The critical module of matrices 9

and

Z ′k,l(a, b) :=

fk+1 bfk 0l−4 −b3fk−2 −b2fk−1

b 0

Tl−2(a, b)

0 b

−fk −bfk−1 0l−4 bfk−1 fk

∈Ml(A).

Also, let

Pk,l(a, b) :=

xk yk 0

−b fk 0

0 0

0 −fk−1 Il−2

and Qk,l(a, b) := I1 ⊕

Il−3 bfk−1 0

0 0

0 fk −b0 yk xk

.

Since

Pk,l(a, b) · Zk,l(a, b) =

1 ∗ ∗ ∗ ∗0 fk+1 bfk 0l−5 −b3fk−2 −b2fk−1

0 b 0

0 Tl−3(a, b)

0 0 b

0 −fk −bfk−1 0l−5 bfk−1 fk

∼u I1 ⊕ Z ′k,l−1(a, b)

for all l ≥ 5,

Qk,l(a, b) · (I1 ⊕ Z ′k,l−1(a, b)) = I1 ⊕

fk+1 bfk 0l−6 b2fk−1 bfk 0

b 0

Tl−4(a, b)

0 b

bfk b2fk−1 0l−6 bfk fk+1 0

∗ ∗ ∗ ∗ 1

for all l − 1 ≥ 5, and det(Pk,l(a, b)) = det(Qk,l(a, b)) = 1, then

Zk,l(a, b) ∼u I1 ⊕ Zk+1,l−2(a, b)⊕ I1 for all l ≥ 6.

Moreover, since Z1,n(a, b) = Cn(a, b), then

Cn(a, b) ∼u

Zq,4(a, b) if n− 2 = 2q,

Z ′q,4(a, b) if n− 2 = 2q + 1.


Finally, using similar reductions we get that

Zq,4(a, b) ∼u

fq bfq−1 b2fq−2 bfq−1

b a b 0

0 b a b

0 −fq 0 fq

∼u

1 0 0 0

0 fq+1 bfq − b3fq−2 −b2fq−1

0 b a b

0 −fq 0 fq

∼u

1 0 0 0

0 afq 2bfq 0

0 b a b

0 −fq 0 fq

∼u

1 0 0 0

0 afq 2bfq 0

0 2bfq afq 0

0 0 0 1

∼u I2 ⊕ fq

(a 2b

2b a

)

and

Z ′q,4(a, b) ∼u

fq+1 bfq + b2fq−1 abfq−1 − b3fq−2 0

b a b 0

0 b a b

−fq −bfq−1 bfq−1 fq

∼u

fq+1 bfq + b2fq−1 bfq 0

b a b 0

bfq bfq + b2fq−1 fq+1 0

0 0 0 1

∼u

1 0 0 0

0 afq+1 − b2fq − b3fq−1 bfq+1 − b2fq 0

0 bfq + b2fq−1 − afq fq+1 − bfq 0

0 0 0 1

∼u I2 ⊕(fq+1 − bfq

)( a+ b b

−1 1

)

∼u I2 ⊕(fq+1 − bfq

)( 1 0

0 a+ 2b

).

Remark. In [17, pag. 44], a simpler proof of Theorem 1.1 (iii) when A is a principal ideal domain can

be found .

The next Lemma give us some useful properties of the polynomials fn(x, y).

Lemma 1.4. If n ≥ 1, then

(i) fn(x, y) =∑bn

2c

i=0 (−1)i(n−ii

)xn−2iy2i,

(ii) fn(x+ y,−1)− fn(y,−1) = x∑n−1

i=0 fi(x+ y,−1)fn−i(y,−1),

(iii) fn(x, y) = fk(x, y)fn−k(x, y)− y2fk−1(x, y)fn−k−1(x, y),

(iv) xkfn(x, y) =∑k

i=0

(ki

)x2ifn+k−2i(x, y).

Proof. It follows using induction on n.

Remark. Note that, fn(x, 0) = xn.

If A is a principal ideal domain and a, b ∈ A, then the equation ax+ by = 1 has a solution if and only

if gcd(a, b) = 1.

Section 2 : Some applications 11

Corollary 1.5. Let A be a principal ideal domain, a, b ∈ A with r = gcd(a, b), a = ra′, and b = rb′.

Then

(i) Tn(a, b) ∼u diag(r, . . . , r, rfn(a′, b′)) for all n ≥ 2,

(ii) Pn(a, b) ∼u diag(r, . . . , r, (a+ 2b)fn−1(a′, b′)) for all n ≥ 2,

(iii) Kn(a, b) ∼u diag(r, a, . . . , a, a′(a+ nb)) for all n ≥ 2, and

(iv) Cn(a, b) ∼u rIn−2 ⊕ C for all n ≥ 4, where

C =

fq(a

′, b′)

a 2b

2b a

if n− 2 = 2q,

(fq+1(a′, b′)− b′fq(a′, b′)

) r 0

0 a+ 2b

if n− 2 = 2q + 1.

Proof. Let Xn(a, b) be either one of the matrices Tn(a, b), Pn(a, b), Kn(a, b), or Cn(a, b), then Xn(a, b) =

rXn(a′, b′). On the other hand, since r = gcd(a, b) if and only if 1 = gcd(a′, b′) if and only if the equation

a′x+ b′y = 1 has solution in A. Then, we get the result applying Theorem 1.1 to Xn(a′, b′).

2. Some applications

In this section we will apply the equivalences of the matrices Kn(a, b), Cn(a, b), Pn(a, b) and Tn(a, b)

obtained in the previous section to the cases when a and b are in the ring of integers and in the subring

Kn(A) of matrices of the form Kn(a, b) where a, b are in a commutative ring with identity A.

At this point we need to introduce some definitions. Given a simple graph G and a natural number

l ≥ 1, the l-duplication of G, denoted by G(l), is the multigraph obtained from G when we replace

every edge of G by l parallel edges. Note that L(G(l)) = lL(G) for any graph G.

v1v2

v3 v4

C4

v1v2

v3 v4

C4(2)

Figure I.1: The simple cycle C4 and its 2-duplication.

Given a graph G and a natural number k, the k-cone of G, denoted by ck(G), is the multigraph obtained

from G by adding a new vertex s and adding k parallel edges between s and all the vertices of G. Note

that L(ck(G), s) = L(G) + kI|V (G)| for any graph G.

The most direct application of the results obtained in the first section is when the base ring A is the

ring of integers and the matrices are the reduced Laplacian matrix of the n-cone of the thick path, the

thick cycle and the thick complete graph.


v1v2

v3 v4C4

s

v1v2

v3 v4c1(C4)

Figure I.2: The cycle C4 and its 1-cone.

s

v1 v2 vn−1 vn

mm m

m

l l

Corollary 2.1. For all m ≥ 0, l ≥ 1, and n ≥ 2, let cm(Pn(l)) be the m-cone of the thick path with

all the edges with multiplicities equal to l. Then

K(cm(Pn(l))) = Zn−1r ⊕ Zmfn−1(m+2l,−l)/rn−1 ,

where r = gcd(l,m).

Proof. Since the reduced Laplacian matrix of cm(Pn(l)), L(cm(Pn(l)), s), is equal to Pn(m + 2l,−l)and gcd(m+ 2l,−l) = gcd(l,m) = r, then by Corollary 1.5 (ii) we get the result.

Corollary 2.2. For all m ≥ 0, l ≥ 1, and n ≥ 4, let cm(Cn(l)) be the m-cone of the thick cycle where

s

v3

v2

v1vn

vn−1

vn−2

m

m

mm

m

m l

l

l

l

l

all the edges has multiplicity equal to l. Then

K(cm(Cn(l))) =

Zn−2r ⊕ Zrsq ⊕ Zmsq if n− 2 = 2q + 1,

Zn−2r ⊕ Zrtq ⊕ Zm(m+4l)tq/r if n− 2 = 2q and m/r is odd,

Zn−2r ⊕ Z2rtq ⊕ Zm(m+4l)tq/2r if n− 2 = 2q and m/r is even,

where r = gcd(l,m), sq = (fq+1(m+ 2l,−l) + lfq(m+ 2l,−l))/rq+1, and tq = fq(m+ 2l,−l)/rq.

Proof. Since L(cm(Cn(l)), s) is equal to Cn(m + 2l,−l) and gcd(m + 2l,−l) = gcd(l,m) = r, then by


Corollary 1.5 (iv), Cn(m+ 2l,−l) ∼u rIn−2 ⊕ C, where

C =

fq(m+ 2l,−l)/rq

m+ 2l −2l

−2l m+ 2l

if n− 2 = 2q,

(fq+1(m+ 2l,−l) + lfq(m+ 2l,−l)

)/rq+1

r 0

0 m

if n− 2 = 2q + 1.

Finally,

(m+ 2l −2l

−2l m+ 2l

)∼u

r 0

0 (m2 + 4ml)/r

if m/r is odd,

2r 0

0 (m2 + 4ml)/2r

if m/r is even.

Corollary 2.3. For all m ≥ 0, l ≥ 1, and n ≥ 4, let cm(Kn(l)) be the m-cone of the thick complete

graph where all the edges have multiplicity equal to l. Then

K(cm(Kn(l))) = Zr ⊕ Zn−2m+nl ⊕ Zm(m+nl)/r,

where r = gcd(l,m).

Proof. Since the reduced Laplacian matrix, L(cm(Kn(l)), s), is equal to Kn(m + nl,−l) and gcd(m +

nl,−l) = gcd(l,m) = r, then by Corollary 1.5 (iii) we get the result.

The subring Kn(A)

In this part we will turn our attention to the case when A is the subring of matrices given by

Kn(A) = {Kn(a, b) | a, b ∈ A} ⊂Mn(A).

At first, we will prove that Kn(A) is a subalgebra of Mn(A).

Lemma 2.4. If a, b, c, d, α ∈ A, then

(i) α ·Kn(a, b) = Kn(α · a, α · b),

(ii) Kn(a, b) +Kn(c, d) = Kn(a+ c, b+ d),

(iii) Kn(a, b) ·Kn(c, d) = Kn(ac, ad+ bc+ nbd),

(iv) Kn(a, b)m = Kn(am, pm(a, b)),


where the polynomials pm,n(x, y) ∈ A[x, y] satisfy the recurrence relation

pnm(x, y) = (x+ ny)pnm−1(x, y) + yxm−1

with initial value pn0 (x, y) = 0.

Proof. The parts (i) and (ii) are straightforward. (iii) Since A(Kn)2 = (n− 1)In + (n− 2)A(Kn), then

Kn(a, b) ·Kn(c, d) = ((a+ b)In + bA(Kn)) · ((c+ d)In + dA(Kn)) = (ac+ ad+ bc+ nbd)In + (ad+ bc+

nbd)A(Kn) = Kn(ac, ad+ bc+ nbd).

(iv) We will use induction on m. The result is clear for m = 1 because pn1 (a, b) = (a+nb)pn0 (a, b)+ba0 =

b. Assume that the result is true for all the natural numbers less or equal to m− 1. Thus

Kn(a, b)m = Kn(a, b)m−1 ·Kn(a, b) = Kn(am−1, pnm−1(a, b)) ·Kn(a, b)

= Kn(am, (a+ nb)pnm−1(a, b) + bam−1) = Kn(am, pnm(a, b)).

Remark. Note that Kn(A) is a commutative ring with identity because Kn(1, 0) = In for all n ∈ N.

On the other hand, since Kn(0, 1)Kn(−n, 1) = 0 = Kn(0, 0), then Kn(A) is not a principal ideal domain

because it has zero divisors.

Remark. Using induction on n is not difficult to see that

pnm(x, y) =m∑i=1

ni−1

(m

i

)xm−iyi.

Also, since pnm(n,−1) = (n− n)pnm−1(n,−1)− (n)m−1 = −nm−1, then

Kn(n,−1)m = nm−1Kn(n,−1) for all m ∈ N.

Now, we will apply Theorem 1.1 (iii) when the base ring is the subring of matrices with entries in

Kn(A) to obtain a theorem that will be a powerful tool to calculate the critical group of several graphs.

Given A = (ai,j), B = (bi,j) ∈Mn(A) and m ≥ 2, let

Φm(A,B) =

Km(a1,1, b1,1) · · · Km(a1,n, b1,n)

.... . .

...

Km(an,1, bn,1) · · · Km(an,n, bn,n)

∈Mn(Km(A)) ⊆Mnm(A).

As the next theorem will show, if a matrix has the block structure of Φm(A,B), then we can get a

simpler equivalent matrix.

Theorem 2.5. Let A = (ai,j), B = (bi,j) ∈Mn(A) and n ≥ 2, then

Φm(A,B) ∼e

[m−2⊕i=1

A

]⊕

(A B

0 A+mB

)


Proof. Let Pn and Qn be as in Claim 1.3, then Im ⊗ Pn, Im ⊗Qn ∈ Mnm(A) are elementary matrices

and

(Im ⊗ Pn) · Φm(A,B) · (Im ⊗Qn) =

Dm(a1,1, b1,1) · · · Dm(a1,n, b1,n)

.... . .

...

Dm(an,1, bn,1) · · · Dm(an,n, bn,n)

∼e

[m−2⊕i=1

A

]⊕

(A B

0 A+mB

),

where Dm(ai,j , bi,j) =

(ai,j bi,j

−mai,j ai,j

)⊕ ai,jIm−2.

In the last part of this paper, we will use Theorem 2.5 in order to find the critical group of some

graphs whose Laplacian matrix is given by Φm(A,B) for some A,B ∈ Mn(A). The simplest case is

when n = 2. If Φm(A,B) is the Laplacian matrix of a graph, then the vertex set of the graph can be

partitioned in two sets and the incidence structure between these sets is given by a matrix in Km(Z).

In this sense, we will define the following families of graphs:

Let U = {u1, u2, . . . , um}, V = {v1, v2, . . . , vm}, and Xy,z be the graph with U ∪ V as vertex set and

edge set equal to Ey ∪ Ez ∪ Ey,z, where

Ey =

∅ if y = m,

uiuj for all i 6= j ∈ {1, 2, · · · ,m} if y = M,

similarly for Ez, and

Ey,z =

uiu′j for all i, j ∈ {1, 2, · · · ,m} if X = K,

uiu′j for all i 6= j ∈ {1, 2, · · · ,m} if X = L,

uiu′i for all i ∈ {1, 2, · · · ,m} if X =M.

Note that Km,m is the bipartite complete graph with m vertices in each partition and Lm,m is the

bipartite complete graph with 2m vertices minus a matching. The Laplacian matrix of all these graphs

can be represented by Φm(A,B) for some two by two matrices A and B.

For instance, the graph MM,m is illustrated in Figure I.3.

KmEy,z

Ey Ez

The trivial graph

Figure I.3: The graph MM,m.

Before using Theorem 2.5 in order to calculate the critical group of the n-cones of graphs Xy,z, we will

introduce a theorem that gives an equivalent matrix of Xy,z when the base ring is a general commutative

ring with identity.


Corollary 2.6. Let m ≥ 2, a, b ∈ A such that the equation ax + by = 1 has a solution in A, and

Xy,y(a, b) = aI2m + bA(Xy,y). Then

(i) Km,m(a, b) ∼u I2 ⊕ aI2(m−2) ⊕ a

(a mb

mb a

),

(ii) Lm,m(a, b) ∼u Im ⊕ (a2 − b2)Im−2 ⊕

(a2 (m− 2)ab

(m− 2)ab a2 − (m− 1)b2

),

(iii) LM,M (a, b) ∼u Im−1 ⊕ a(a− 2b)Im−2 ⊕

a(a− 2b) ab 0

0 a+ 2(m− 1)b 0

0 (m− 1)b a

,

(iv) MM,M (a, b) ∼u Im+1 ⊕ a(a− 2b)Im−2 ⊕

(a(a− 2b) −b2(2a+ (m− 2)b)

0 (a+ (m− 1)b)2 − b2

).

Proof. (i) Since Km,m(a, b) = Φm(A,B) for A =

(a 0

0 a

)and B =

(0 b

b 0

), then by Theorem 2.5

Km,m(a, b) ∼e

[m−2⊕i=1

(a 0

0 a

)]⊕

a 0 0 b

0 a b 0

0 0 a mb

0 0 mb a

.

Moreover, sincea 0 0 b

0 a b 0

0 0 a mb

0 0 mb a

x 0 0 −b0 1 0 0

0 0 1 0

y 0 0 a

=

1 0 0 0

0 a b 0

mby 0 a mab

ay 0 mb a2

∼e

1 0 0 0

0 a b 0

0 0 a mab

0 0 mb a2

and a b 0

0 a mab

0 mb a2

x −b 0

y a 0

0 0 1

=

1 0 0

ay a2 mab

mby mab a2

∼e 1 0 0

0 a2 mab

0 mab a2

,

then a 0 0 b

0 a b 0

0 0 a mb

0 0 mb a

∼u I2 ⊕ a

(a mb

mb a

).

Hence

Km,m(a, b) ∼u I2 ⊕ aI2(m−2) ⊕ a

(a mb

mb a

).


(ii) Since Lm,m(a, b) = Φm(A,B) for A =

(a −b−b a

)and B =

(0 b

b 0


Lm,m(a, b) ∼e

[m−2⊕i=1

(a −b−b a

)]⊕

a −b 0 b

−b a b 0

0 0 a (m− 1)b

0 0 (m− 1)b a

.

Moreover, since(a −b−b a

)(x b

−y a

)=

(1 0

−by − ay a2 − b2

)∼e

(1 0

0 a2 − b2

),

a −b 0 b

−b a b 0

0 0 a (m− 1)b

0 0 (m− 1)b a

∼e

a 0 0 b

−b a b 0

0 (m− 1)b a (m− 1)b

0 a (m− 1)b a

,

1 0 0 0

∗ a b b2

∗ (m− 1)b a (m− 1)ab

∗ a (m− 1)b a2

=

a 0 0 b

−b a b 0

0 (m− 1)b a (m− 1)b

0 a (m− 1)b a

x 0 0 −b

0 1 0 0

0 0 1 0

y 0 0 a

∼e I1 ⊕

a b 0

(m− 1)b a (m− 2)ab

a (m− 1)b a2 − (m− 1)b2

,

and 1 0 0

∗ a2 (m− 2)ab

∗ (m− 2)ab a2 − (m− 1)b2

=

a b 0

(m− 1)b a (m− 2)ab

a (m− 1)b a2 − (m− 1)b2

x −b 0

y a 0

0 0 1

∼e I1 ⊕

(a2 (m− 2)ab

(m− 2)ab a2 − (m− 1)b2

),

then

Lm,m(a, b) ∼u Im ⊕ (a2 − b2)Im−2 ⊕

(a2 (m− 2)ab

(m− 2)ab a2 − (m− 1)b2

).

(iii) Since LM,M (a, b) = Φm(A,B) for A =

(a− b −b−b a− b

)and B =

(b b

b b


LM,M (a, b) ∼e

[m−2⊕i=1

(a− b −b−b a− b

)]⊕

a− b −b b b

−b a− b b b

0 0 a+ (m− 1)b (m− 1)b

0 0 (m− 1)b a+ (m− 1)b

.


Moreover, since(x −(x+ y)

b a− b

)(a− b −b−b a− b

)=

(1 ∗0 a(a− 2b)

)∼e

(1 0

0 a(a− 2b)

),

x −(x + y) 0

b a− b 0

0 0 I2

a− b −b b b

−b a− b b b

0 0 a + (m− 1)b (m− 1)b

0 0 (m− 1)b a + (m− 1)b

=

1 ∗ ∗ ∗0 a(a− 2b) ab ab

0 0 a + (m− 1)b (m− 1)b

0 0 (m− 1)b a + (m− 1)b

,

and a(a− 2b) ab ab

0 a+ (m− 1)b (m− 1)b

0 (m− 1)b a+ (m− 1)b

∼e a(a− 2b) ab 0

0 a+ 2(m− 1)b 0

0 (m− 1)b a

then the result is followed.

(iv) Since MM,M (a, b) = Φm(A,B) for A =

(a− b b

b a− b

)and B =

(b 0

0 b


MM,M (a, b) ∼e

[m−2⊕i=1

(a− b b

b a− b

)]⊕

a− b b b 0

b a− b 0 b

0 0 a+ (m− 1)b b

0 0 b a+ (m− 1)b

.

Moreover, since

x x + y 0

−b a− b 0

0 0 I2

a− b b b 0

b a− b 0 b

0 0 a + (m− 1)b b

0 0 b a + (m− 1)b

=

1 ∗ ∗ ∗0 a(a− 2b) −b2 (a− b)b0 0 a + (m− 1)b b

0 0 b a + (m− 1)b

and a(a− 2b) −b2 (a− b)b

0 a + (m− 1)b b

0 b a + (m− 1)b

1 0 0

0 a + (m− 1)b y − (m− 1)x

0 −b x

=

a(a− 2b) −b2(2a + (m− 2)b) ∗0 (a + (m− 1)b)2 − b2 ∗0 0 1

,

then the result is followed.

Remark. Note that KM,M is the complete graph with 2m vertices and Mm,m is the disjoint union of

m copies of K2.

When a graph G is not regular, then there is not a straightforward way to define their matrix G(a, b).

Thus, we will define Km,M (a, b) as a(Im ⊕ 2Im) + bA(Km,M ), Lm,M (a, b) as a(Im ⊕ 2Im) + bA(Lm,M ),

and Mm,M (a, b) as a(Im ⊕ (m + 1)Im) + bA(Mm,M ). Now, we have the following equivalent matrices

of Km,M (a, b), Lm,M (a, b), and Mm,M (a, b).

Corollary 2.7. Let m ≥ 2, a, b ∈ A such that the equation ax+ by = 1 has a solution in A, then

(i) Km,M (a, b) ∼u I2 ⊕ aIm−2 ⊕ 2aIm−2 ⊕ a

(2a −(a−mb)b2m 2a

),


(ii) Mm,M (a, b) ∼u Im ⊕ ((m+ 1)a2 − b2)Im−2 ⊕

((m+ 1)a2 − b2 b

0 (a+ b)(b− (m+ 1)a)

),

(iii) Lm,M (a, b) ∼u Im−1 ⊕ (2a2 − b2)Im−2 ⊕

2a2 − b2 0 ab+mb2

0 a (m− 1)b

0 (m− 1)b 2a+mb

.

Proof. (i) Since Km,M (a, b)=Φm(A,B) for A =

(a 0

0 2a

)and B =

(0 b

b b


Km,M (a, b) ∼e

[m−2⊕(

a 0

0 2a

)]⊕

a 0 0 b

0 2a b b

0 0 a mb

0 0 mb 2a+mb

.

Moreover,a 0 0 b

0 2a b b

0 0 a mb

0 0 mb 2a+mb

x 0 0 −b0 1 0 0

0 0 1 0

y 0 0 a

=

1 0 0 0

∗ 2a b ab

∗ 0 a mab

∗ 0 mb 2a2 +mab

∼e

1 0 0 0

0 b 2a ab

0 a 0 mab

0 0 −2ma 2a2

and y x 0

−a b 0

0 0 1

b 2a ab

a 0 mab

0 −2ma 2a2

=

1 ∗ ∗0 −2a2 mab2 − a2b

0 −2ma 2a2

∼e I1⊕

(2a2 ab(mb− a)

2ma 2a2

).

(ii) Since Mm,M (a, b) = Φm(A,B) for A =

(a b

b (m+ 1)a

)and B =

(0 0

0 b


Mm,M (a, b) ∼

[m−2⊕(

a b

b (m+ 1)a

)]⊕

a b 0 0

b (m+ 1)a 0 b

0 0 a b

0 0 b (m+ 1)a+mb

Moreover,(

a b

b (m+ 1)a

)(x −by a

)=

(1 0

∗ (m+ 1)a2 − b2

)∼e

(1 0

0 (m+ 1)a2 − b2

)


anda b 0 0

b (m+ 1)a 0 b

0 0 a b

0 0 b (m+ 1)a+mb

∼u

1 0 0 0

∗ (m+ 1)a2 − b2 0 b

∗ 0 a b

∗ 0 b (m+ 1)a+mb

∼e I1 ⊕

b (m+ 1)a+mb 0

a b 0

0 b (m+ 1)a2 − b2

∼u I2 ⊕

((a+ b)(b− (m+ 1)a) 0

b (m+ 1)a2 − b2

)

(iii) Since Lm,M (a, b) = Φm(A,B) for A =

(a −b−b 2a

)and B =

(0 b

b b


Lm,M (a, b) ∼

[m−2⊕(

a −b−b 2a

)]⊕

a −b 0 b

−b 2a b b

0 0 a (m− 1)b

0 0 (m− 1)b 2a+mb

.

Moreover, (x −yb a

)(a −b−b 2a

)=

(1 ∗0 2a2 − b2

)∼e

(1 0

0 2a2 − b2

),

a −b 0 b

−b 2a b b

0 0 a (m− 1)b

0 0 (m− 1)b 2a+mb

∼u

1 0 0 0

0 2a2 − b2 ab ab+ b2

0 0 a (m− 1)b

0 0 (m− 1)b 2a+mb

,

and 2a2 − b2 ab ab+ b2

0 a (m− 1)b

0 (m− 1)b 2a+mb

∼e

2a2 − b2 0 ab+mb2

0 a (m− 1)b

0 (m− 1)b 2a+mb

.

The next corollaries calculate the critical group of the n-cone of the graphs Xy,z.

Corollary 2.8. Let m ≥ 2, l ≥ 1, and n ≥ 0, then

K(cn(Km,m(l))) = Z2r ⊕ Z2(m−2)

n+ml ⊕ Z(n+ml)s/r ⊕ Zn(n+ml)(n+2ml)/rs,

where r = gcd(l, n) and s = gcd(ml, n).


Proof. Since L(cn(Km,m(l)), s) = Km,m(n+ml,−l) and Km,m(n+ml,−l) = rKm,m((n+ml)/r,−l/r)with gcd((n+ml)/r,−l/r) = 1, then applying Corollary 2.7 (i) to Km,m((n+ml)/r,−l/r) we get that

Km,m(n+ml,−l) ∼u rI2 ⊕ (n+ml)I2(m−2) ⊕ ((n+ml)/r)

(n+ml −ml−ml n+ml

).

On the other hand, (n+ml −ml−ml n+ml

)∼u

(s 0

0 n(n+ 2ml)/s

),

where s = gcd(ml, n) and we get the result.

Remark. Note that Km,m is the complete bipartite graph with m vertices in each partition. Lorenzini

in [31] calculated that

K(Km,m) = Z2(m−2)m ⊕ Zm2 ,

which agrees with the Corollary 2.8 for l = 1 and n = 0. Also note that K(cn(Km,m(l))) has 2m − 2

invariant factors different to 1.


K(cn(Lm,m(l))) = Zmr ⊕ Zm−2(s2−l2)/r

⊕ Zrt ⊕ Zu/r3t,

where r = gcd(l, n), s = n+ (m− 1)l, t = gcd(m− 1, n)/gcd(l,m− 1, n), and u = s2(n2 + 2n(m− 1)l+

(m− 2)l2).

Proof. In a similar way that in Corollary 2.8, L(cn(Lm,m(l)), s) = Lm,m(n+ (m−1)l,−l) and applying

Corollary 2.7 (ii) to Lm,m(a/r, b/r) with a = n+ (m− 1)l and b = −l

Lm,m(n+ (m− 1)l,−l) ∼u rIm ⊕ (s2 − l2)/rIm−2 ⊕

(a2/r (m− 2)ab/r

(m− 2)ab/r (a2 − (m− 1)b2)/r

),

where s = n+ (m− 1)l.

On the other hand, it is not difficult to see that(a2/r (m− 2)ab/r

(m− 2)ab/r (a2 − (m− 1)b2)/r

)∼u

(rt 0

0 u/r3t

),

where t = gcd(m− 1, n)/gcd(l,m− 1, n) and u = s2(n2 + 2n(m− 1)l + (m− 2)l2).


K(cn(LM,M (l))) = Zm−1r ⊕ Zm−2

st/r ⊕ Zu ⊕ Zsv/u ⊕ Znst/rv,

where r = gcd(l, n), s = n+2(m−1)l, t = n+2ml, u = gcd(n, (m−1)l), and v = gcd(n, 2(m−1)l2/r).


Proof. Since L(cn(LM,M (l)), s) = LM,M (n+2(m−1)l,−l) and r = gcd(n+2(m−1)l,−l), then applying

Corollary 2.7 (iii) to LM,M (a/r, b/r) with a = n+ 2(m− 1)l and b = −l we get that

LM,M (n+ 2(m− 1)l,−l) ∼u rIm−1 ⊕ st/rIm−2 ⊕

st/r −sl/r 0

0 n 0

0 −(m− 1)l s

,

where r = gcd(l, n), s = n+ 2(m− 1)l, and t = n+ 2ml.

On the other hand, it is not difficult to see that a(a− 2b)/r ab/r 0

0 a+ 2(m− 1)b 0

0 (m− 1)b a

=

st/r −sl/r 0

0 n 0

0 −(m− 1)l s

∼u u⊕ sv/u⊕ nst/rv,where u = gcd(n, (m− 1)l) and v = gcd(n, 2(m− 1)l2/r).


K(cn(MM,M (l))) = Zm+1r ⊕ Zm−2

(n+ml)(n+(m+2)l)/r ⊕ Zu ⊕ Zn(n+2l)(n+ml)(n+(m+2)l)/ur2 ,

where r = gcd(l, n), u = gcd(n(n+ 2l), lv(n+ t))/r, and v = gcd(m, l/r).

Proof. Since L(cn(MM,M (l)), s) = MM,M (n + ml,−l) and r = gcd(n + ml,−l), then applying Corol-

lary 2.7 (iii) to MM,M (a/r, b/r) with a = n+ml and b = −l we get that

MM,M (n+ml,−l) ∼u rIm+1 ⊕ st/rIm−2 ⊕

(st/r −l2(n+ t)/r2

0 n(n+ 2l)/r,

)

where r = gcd(l, n), s = n+ml, and t = n+ (m+ 2)l.

On the other hand, it is not difficult to see that(st/r −l2(n+ t)/r2

0 n(n+ 2l)/r

)∼u u⊕ stn(n+ 2l)/ur2,

where u = gcd(n(n+ 2l), lv(n+ t))/r and v = gcd(m, l/r).

Remark. Note that MM,M (l) is the cartesian product of K2(l) and Kn(l). A deep analysis of the

cartesian product of matrices can be found in [19].


K(cn(Km,M (l))) = Zm−2ml+n ⊕ Zm−2

2ml+n ⊕ Z2r ⊕ Zs(n+2ml)/r2 ⊕ Zn(n+ml)(n+2ml)/s,

where r = gcd(l, n) and s = gcd(n2,mlr).


Proof. Since L(cn(Km,M (l))) = Φm(A,B) for A =

(ml + n 0

0 2ml + n

)and B =

(0 −l−l −l

), then

the result is followed by Theorem 2.7 (iv) becauseml + n 0 0 −l

0 2ml + n −l −l0 0 ml + n −ml0 0 −ml ml + n

∼u rI2 ⊕ (n+ 2ml)s/r2 ⊕ n(n+ml)(n+ 2ml)/s,

where r = gcd(l, n) and s = gcd(n2,mlr).

Remark. Note that Km,M is the graph K2m \ Km. In general the expression for K(cn(Km,M (l)))

given in Corollary 2.12 does not give us the invariant factors of K(cn(Km,M (l))). Also note that

K(cn(Km,M (l))) has 2m− 2 invariant factors different to 1.


K(cn(Mm,M (l))) = Zm+1r ⊕ Zm−2

s/r ⊕ Zn(n+2l)s/r3 ,

where r = gcd(l, n) and s = n2 +ml2 + nl(m+ 2).

Proof. Since L(cn(Mm,M (l))) = Φm(A,B) for A =

(l + n −l−l (m+ 1)l + n

)and B =

(0 0

0 −l

),

then the result is followed by Theorem 2.7 (v) because(l + n −l−l (m+ 1)l + n

)∼u r ⊕ s/r

and l + n −l 0 0

−l (m+ 1)l + n 0 −l0 0 l + n −l0 0 −l l + n

∼u rI3 ⊕ n(n+ 2l)s/r3,

where r = gcd(l, n) and s = n2 +ml2 + nl(m+ 2).

We conclude the chapter with the critical group of the graph Lm,M .


K(Lm,M ) = Zmr ⊕ Zm−2s/r ⊕ Zt ⊕ Zn(n+2(m−1)l)s/tr2 ,

where r = gcd(l, n), s = n2 + (3m− 2)nl +m(2m− 3)l2, and t = gcd(n, l3(m− 1)(2m− 3)/r2).

Proof. Since L(cn(Lm,M (l))) = Φm(A,B) for

A =

((m− 1)l + n l

l (2m− 1)l + n

)and B =

(0 −l−l −l

),


then the result is followed by Theorem 2.7 (vi) because((m− 1)l + n l

l (2m− 1)l + n

)∼ r ⊕ s/r,

where r = gcd(l, n), s = n2 + (3m− 2)nl +m(2m− 3)l2, and(m− 1)l + n l 0 −l

l (2m− 1)l + n −l −l0 0 (m− 1)l + n −(m− 1)l

0 0 −(m− 1)l (m− 1)l + n

∼u rI2 ⊕ t⊕ n(n+ 2(m− 1)l)s/tr2,

where t = gcd(n, l3(m− 1)(2m− 3)/r2).

Chapter II : Critical ideals of graphs

If XG = {xu |u ∈ V (G)} is the set of variables indexed by the vertices of the graph G, then the

generalized Laplacian matrix of G, denoted by L(G,XG), is given by

L(G,XG)u,v =

xu if u = v,

−m(u,v) otherwise.

Furthermore, if P is a commutative ring with identity, P[XG] is the polynomial ring over P in the

variables XG and 1 ≤ i ≤ n, then the i-critical ideal of G is the determinantal ideal given by

Ii(G,XG) = 〈minorsi(L(G,XG))〉 ⊆ P[XG],

where minorsi(L(G,XG)) is the set of determinants of all the i-square submatrices of L(G,XG). Note

that we can define (without any technical issue) the critical ideals of an n× n matrix M with entries

in P as Ii(M,X) = 〈minorsi(L(M,X))〉 ⊆ P[X] for all i = 1, . . . , n, where

L(M,X)u,v =

xu if u = v,

−Mu,v otherwise.

The critical ideals and the critical group of a graph are closely related, as Propositions 1.6 and 1.7 show.

Moreover, critical ideals of a graph are very useful to get a better understanding of its critical group. In

Section 1, we will show that the critical ideals are better behaved than the critical group. For instance,

if γP(G) is the number of critical ideals over the base ring P that are trivial, then Theorem 1.12 asserts

that

γP(G) ≤ 2(|V (G)| − α(G)) and γP(G) ≤ 2(|V (G)| − ω(G)) + 1,

where α(G) is the stability number and ω(G) is the clique number of the graph. That is; the invariant

γP is closely related to the combinatorics of the graph. Also, if H is an induced subdigraph of G, then

γP(H) ≤ γP(G) in contrast with the behavior of the number of invariant factors equal to one of the

critical group of induced subdigraphs.

The main goal of this chapter is the study of the critical ideals of graphs. The main results of this

chapter are contained in Sections 1 and 2. Section 1 contains the basic properties of the critical ideals

26 Chapter II : Critical ideals of graphs

and we define the invariant γP as the number of the critical ideals that are trivial. Also, in this section

we get the reduced Grobner basis of the complete graph. Finally, we explore the relationship of the

critical ideals of a graph with the characteristic polynomial of its adjacency and Laplacian matrices.

The reduced Grobner basis of the critical ideals of the cycles and some combinatorial expression for

the minors of the generalized Laplacian matrix of a digraph are presented in Section 2.

1. Critical ideals of graphs

In this section, we introduce the main concept of this chapter: the critical ideals of a digraph G. We will

begin this section by defining the critical ideals of a digraph, presenting some examples and discussing

some of their basic properties. In general terms, the i-th critical ideal is the determinantal ideal of

i-minors of the generalized Laplacian matrix associated to G. The critical ideals of G generalize the

critical group of G (see Proposition 1.6) and the characteristic polynomials of the adjacency matrix

and the Laplacian matrix associated to G (see Section ). Moreover, with some additional requirements

over G we can get a stronger correspondence between the critical ideals of G and the critical group of

G, see Proposition 1.7.

Afterward, we introduce the number of critical ideals that are trivial as an invariant of the digraph.

In the case of graphs, we will establish a bound between this invariant and the stability and clique

numbers of the graph. Also, in Section we present a minimal set of generators and a reduced Grobner

basis for the critical ideals of the complete graphs. As a byproduct we will get expressions for the

critical groups for a complete graph minus a star. Finally, we will explore the relation between the

critical ideals of a graph and the characteristic polynomial of its adjacency and Laplacian matrix.

Definition 1.1. Given a digraph G with n vertices and 1 ≤ i ≤ n, let

Ii(G,X) = 〈minorsi(L(G,X))〉 ⊆ P[XG]

be the i-th critical ideal of G.

Note that in general the critical ideals depend on the base ring P, in this chapter we are mainly

interested when P = Z. By convention, Ii(G,X) = 〈1〉 if i ≤ 0 and Ii(G,X) = 〈0〉 if i > n. Clearly

In(G,X) is a principal ideal generated by the determinant of the generalized Laplacian matrix.

Now, we present an example that illustrates the concept of critical ideal.

Example 1.2. Let H be the complete graph with six vertices minus the perfect matching formed by

the edges M3 = {v1v4, v2v5, v3v6} (see Figure II.1(a)) and P = Z. Then,

L(H,X) =

x1 −1 −1 0 −1 −1

−1 x2 −1 −1 0 −1

−1 −1 x3 −1 −1 0

0 −1 −1 x4 −1 −1

−1 0 −1 −1 x5 −1

−1 −1 0 −1 −1 x6

Section 1 : Critical ideals of graphs 27

Using any algebraic system, for instance Macaulay 2, it is not difficult to see that Ii(H,X) = 〈1〉 for

i = 1, 2 and

Ii(H,X) =

⟨2, x1, x2, x3, x4, x5, x6

⟩if i = 3,⟨

{xrxs | vrvs ∈ E(H)} ∪ {2xr + 2xs + xrxs | vrvs 6∈ E(H)}⟩

if i = 4,⟨{xkxl(xr + xs + xrxs) | (r, s, k, l) ∈ S(H)} ∪ {p(r,s,k,l) | vrvs, vkvl 6∈ E(H)}

⟩if i = 5,⟨

x1x2x3x4x5x6 −∑

(r,s,k,l)∈S(H) xrxsxkxl − 2∑

(r,s,k)∈T (H) xrxsxk⟩

if i = 6,

where

S(H) = {(r, s, k, l) | vrvs 6∈ E(H), vkvl ∈ E(H), and {i, j} ∩ {k, l} = ∅},

T (H) are the triangles of H, and p(r,s,k,l) = (xr + xs)(xk + xl + xkxl) + (xk + xl)(xr + xs + xrxs). Note

that the expressions of the critical ideals of H depend heavily on their combinatorics.

Now, let us turn our attention to one of the most basic properties of the critical ideals.

Proposition 1.3. If G is a digraph with n vertices, then

〈0〉 ( In(G,X) ⊆ · · · ⊆ I2(G,X) ⊆ I1(G,X) ⊆ 〈1〉.

Moreover, if H is an induced subdigraph of G with m vertices, then Ik(H,X) ⊆ Ik(G,X) for all 1 ≤k ≤ m.

Proof. Let M be a (k + 1) × (k + 1) matrix over P[XG]. Since det(M) =∑k+1

i=1 Mi,1 det(M(i; 1)),

Ik+1(G,X) ⊆ Ik(G,X) for all 1 ≤ k ≤ n − 1. On the other hand, since any submatrix of L(H,X) is

also a submatrix of L(G,X), minorsk(L(H,X)) ⊆ minorsk(L(G,X)) for all 1 ≤ k ≤ m and therefore

Ik(H,X) ⊆ Ik(G,X) for all 1 ≤ k ≤ m.

v5

v4v1

v2

v3 v6

v5

v4v1

v2

v3 v6

v

v1

v2 v3

v4

v5 v6

(a) K6 \M3 (b) G1 (c) G2

Figure II.1: (a) K6 \M3, (b) G1, (c) G2.

If the digraph is not connected, then we can express its critical ideals as a function of the critical ideals

of its connected components.

Proposition 1.4. If G and H are vertex disjoint digraphs, then

Ii(G tH,X) =

⟨i⋃

j=0

Ij(G,X)Ii−j(H,X)

⟩for all 1 ≤ i ≤ |V (G tH)|.


Proof. Let Q = M ⊕ N where M ∈ Mm(P[X]) and N ∈ Mn(P[X]), k ∈ [m + n], and r, s ⊆ [m + n]

with |r| = |s| = k. It follows by using induction on |rm|, that

det(Q[r; s]) =

det(Q[rm; sm]) · det(Q[rn; sn]) if |rm| = |sm|,

0 otherwise,

where rm = r ∩ [m], sm = s ∩ [m], rn = r \ rm, and sn = s \ sm.

Now, since L(GtH,X) = L(G,X)⊕L(H,X), we get that minorsi(L(GtH,X)) \ 0 ⊆ {m1 ·m2 |m1 ∈minorsj(L(G,X)) and m2 ∈ minorsi−j(L(H,X)) for some 0 ≤ j ≤ i} for all 1 ≤ i ≤ |V (G ∪H)| and

the result is obtained.

Let Tv be the trivial graph composed by the vertex v. Since I1(Tv) = 〈xv〉, applying Proposition 1.4,

we get the critical ideals of the trivial graph with n vertices.

Corollary 1.5. If n ≥ 1 and Tn is the graph with n isolated vertices, then

Ii(Tn) =⟨{∏j∈J

xj∣∣ |J | = i}

⟩for all 1 ≤ i ≤ n.

Now, we will establish some basic relationships between the critical ideals and the critical group. Before

doing this, we will introduce some notation. Given a digraph G with n vertices and d ∈ PV (G), let

L(G,d) be the matrix obtained from L(G,X) where we put xv = dv for all v ∈ V (G). Also, for all

1 ≤ i ≤ n, let

Ii(G,d) = {f(d) | f ∈ Ii(G,X)} ⊆ Z.

Given an induced subdigraph H of G, the degree vector of H in G is given by dG(H)v = deg+G(v) for

all v ∈ V (H).

Proposition 1.6. Let P = Z and G be a digraph (possibly with multiple edges) with n vertices. If

K(G) ∼=⊕n−1

i=1 Zfi with f1| · · · |fn−1, then

Ii(G, dG(G)) = 〈i∏

j=1

fj〉 for all 1 ≤ i ≤ n− 1.

Proof. Clearly L(G, dG(G)) is equal to the Laplacian matrix L(G) of G. Thus minorsi(L(G, dG(G))) =

minorsi(L(G)) for all 1 ≤ i ≤ n and therefore

Ii(G, dG(G)) = 〈minorsi(L(G, dG(G)))〉 = 〈minorsi(L(G))〉 = 〈i∏

j=1


On the other hand, if v is a vertex of G and L(G, v) is the reduced Laplacian matrix, the matrix

obtained from L(G) by removing the row and column v, then we have the following strong version of

Proposition 1.6:


Proposition 1.7. Let P = Z, G be a connected digraph (possibly with multiple edges) with n vertices,

and v a vertex of G. If G is Eulerian (that is, d+G(v) = d−G(v) for all v ∈ V (G)) and K(G) ∼=

⊕n−1i=1 Zfi

with f1| · · · |fn−1, then

Ii(G \ v, dG(G \ v)) = 〈i∏

j=1


Proof. Since L(G \ v, dG(G \ v)) = L(G, v) (the reduced Laplacian matrix of G), by Proposition 1.6 we

only need to prove that K(G) ∼= ZV (G)\v/ImL(G, v)t. Now, if In−1,n−1 is the identity matrix of size

n− 1, then [1 1

0 In−1,n−1

]L(G)

[1 1

0 In−1,n−1

]t= 0⊕ L(G, v).

Since det(

[1 1

0 In−1,n−1

]) = 1, then L(G) ∼ 0⊕ L(G, v) and we get the result.

Remark. Note that, in general, Proposition 1.7 is not valid for digraphs. However, we can get a

similar result for matrices in Mn×n with entries in a principal ideal domain and such that M1 = 0 and

1M = 0.

The next example shows how Proposition 1.7 can be used to recover the critical group of a graph from

its critical ideals. In this sense, critical ideals generalize the critical group of a graph.

Example 1.8. Let H be the complete graph with six vertices minus a perfect matching as in Fig-

ure II.1(a). and G be a graph such thatH = G\v for some vertex v ofG. Thus, applying Proposition 1.7

we can get the critical group of G as an evaluation of the critical ideals of H. For instance, if G1 is the

graph obtained from H (see Figure II.1(b)) by adding a new vertex v and the edges vv1, vv3, vv4, vv6,

then dG(H) = (5, 4, 5, 5, 4, 5). Moreover, using the critical ideals of H calculated in Example 1.2, we

get that fi = 1 for all i ≤ 4, f5 = 20, f6 = 140; that is,

K(G1) ∼= Z20 ⊕ Z140.

On the other hand, if we only know the critical ideals of induced subgraphs of G that are different to

G \ v, then we cannot reconstruct completely the critical group of G. For instance, if G2 is the graph

obtained by adding the vertices v5, v6 and the edges v5v1, v5v2, v6v2, v6v3, v6v4 to the complete graph

with four vertices (see Figure II.1(c)), then it is not difficult to see using any algebraic system that

fi(G2) = 1 for 1 ≤ i ≤ 4 and f5(G2) = 185. However, when we apply Proposition 1.7 to the critical

ideals of the induced subgraph by the vertices v1, v2, v3, and v4 of G2 (isomorphic to K4) we can only

obtain that f1(G2) = f2(G2) = 1, f3(G2) | 5 and f4(G2) | 175.

The invariant γ

In this subsection, we will present an invariant that will play an important role on the study of the

critical ideals of a digraph.


Definition 1.9. Given a digraph G and a commutative ring with identity P, let

γP(G) = max{i | Ii(G,X) = 〈1〉}.

Using the canonical homomorphisms f : Z → P given by f(a) = af(1) is not difficult to get that

γZ(G) ≤ γP(G). For instance, it is clear that γZ(G) ≤ γQ(G). Also, there exists a close relation

between γZ(G) and the number of invariant factors of the critical group of G that are equal to 1. For

instance, if IF1(G) denote the number of invariant factors of the critical group of G that are equal to

1, then γZ(G) ≤ IF1(G). We found that γP(G) behaves better than the number of invariant factors of

the critical group of a digraph that are one. For instance, it is not difficult to see from the definition

and Proposition 1.3 that if H is an induced subdigraph of G, then γP(H) ≤ γP(G). However, if

n ≥ 3, G = K2,n is a complete bipartite graph, and H = K1,n is an induced subgraph of G, then

IF1(G) = 2 < n = IF1(H).

Now, we present a relation between γP(G) and the stability and the clique numbers of G. Before doing

this, we will define the stability and the clique numbers of a graph. A subset S of the vertices of a

graph G is called stable or independent if there is no edge of G with ends in S. A stable set is called

maximal if it is under the inclusion of sets. The stability number of G, denoted by α(G), is given by

α(G) = max{|S| |S is a stable set of G}.

In a similar way, a subset C of the vertices of a graph G is called a clique if all the pairs of vertices in

C are joined by an edge of G. A clique set is called maximal if it is under the inclusion of sets. The

clique number of G, denoted by ω(G), is given by

ω(G) = max{|C| |C is a clique set of G}.

Lemma 1.10. If G is a digraph (possibly with multiple edges) and v is a vertex of G, then

γP(G)− γP(G \ v) ≤ 2.

Proof. We begin with a simple relation between the critical ideals of G and G \ v.

Claim 1.11. If G is a digraph (possibly with multiple edges) with V (G) = {v1, . . . , vn}, then

Ij(G,X) ⊆ 〈x1Ij−1(G \ v1, X \ x1), Ij−2(G \ v1, X \ x1)〉 for all 1 ≤ j ≤ n.

Proof. Let I = {i1, . . . ij} ⊆ [n], I ′ = {i′i, . . . , i′j} ⊆ [n] and mI,I′ = det(L(G,X)[I, I ′]) ∈ Ij(G). If

1 /∈ I∪I ′, then mI,I′ ∈ Ij(G\v1, X \x1). In a similar way, if 1 ∈ I∆I ′, then mI,I′ ∈ Ij−1(G\v1, X \x1).

On the other hand, if 1 ∈ I ∩ I ′, then

mI,I′ ∈ 〈x1Ij−1(G \ v1, X \ x1), Ij−2(G \ v1, X \ x1)〉.

Finally, the result follows because Ij(G \ v1, X \ x1) ⊆ Ij−1(G \ v1, X \ x1) ⊆ Ij−2(G \ v1, X \ x1).


Let g = γP(G \ v1). Since Ii(G \ v1, X \ x1) 6= 〈1〉 for all i ≥ g + 1, by Claim 1.11

Ig+3(G,X) ⊆ 〈x1Ig+2(G \ v1, X \ x1), Ig+1(G \ v1, X \ x1)〉 6= 〈1〉.

Therefore Ig+3(G,X) 6= 〈1〉. That is, γP(G)− γP(G \ v1) ≤ 2.

Since γP(Tm) = 0 and γP(Kn+1) = 1 for all m,n ≥ 1, then using Lemma 1.10 we have the following

result:

Theorem 1.12. If G is a digraph (possibly with multiple edges) with n vertices, then

γP(G) ≤ 2(n− ω(G)) + 1 and γP(G) ≤ 2(n− α(G)).

Proof. The result follows by using that γP(Tα(G)) = 0 (Corollary 1.5), γP(Kω(G)) = 1 (Theorem 1.15)

and the fact that γP(G)− γP(G \ v) ≤ 2 for all v ∈ V (G) (Lemma 1.10).

This result is interesting when the stability or the clique number is almost the number of vertices of

the graph. For instance, if G is the complete graph (ω(G) = n), then γP(G) ≤ 1. Theorem 1.15 proves

that this bound is tight. Similarly to Theorem 1.12, in [31] an upper bound for the number of invariant

factors different to one of the critical group of a graph G in terms of the number of independent cycles

of G was found.

In [29] a similar result to the obtained in the proof of Theorem 1.12 was obtained. Namely in [29] it

was shown that if G is a simple graph and e ∈ E(G), then the number of invariant factors different to

1 of G and G \ e differ by at most 1.

Clearly, a simple graph G has γP(G) = 0 if and only if G is the trivial graph. Moreover, in [4] it was

shown that a simple graph G has γZ(G) = 1 if and only if G is the complete graph. Also, all the simple

graphs with γZ equal to 2 were characterized in [4].

It is not difficult to prove that the bound γP(G) ≤ 2(n − α(G)) is tight. For instance, it is easy to

prove that if P2n+1 is an odd path (see Corollary 2.8), then α(P2n+1) = n+ 1 and γP(P2n+1) = 2n =

2(2n+ 1− (n+ 1)). Moreover, in Chapter III it was shown that if T is a tree, then γZ(T ) is equal to

its 2-matching number. This result proves that the bound γP(G) ≤ 2(n− α(G)) is tight for any value

of the stability number and the number of vertices of the graph. An interesting open question is the

characterization of the simple graphs that satisfy the bounds given in Theorem 1.12.

Next example shows a graph G with γZ(G) = 5 such that L(G,X) has no 5-minor equal to 1.

Example 1.13. Let G be the cone of H (obtained from H when we add a new vertex v and all the

edges between the vertex v and the vertices of H), see Figure II.2. Since

det(L(G,X)[{1, 2, 3, 4, 5}, {2, 3, 5, 6, 7}]) = x2 + x5 + x2x5 and

det(L(G,X)[{1, 2, 3, 5, 6}, {2, 4, 5, 6, 7}]) = −(1 + x2 + x5 + x2x5),

then γZ(G) = 5. However, it is not difficult to check that no 5-minor of L(G,X) is equal to one or

another integer.

Now, we turn our attention to the critical ideals of the complete graph.


v1

v2 v3 v4 v5

v6H

L(c(H), X) =

x1 −1 −1 0 0 −1 −1

−1 x2 −1 0 0 0 −1

−1 −1 x3 −1 0 −1 −1

0 0 −1 x4 −1 −1 −1

0 0 0 −1 x5 −1 −1

−1 0 −1 −1 −1 x6 −1

−1 −1 −1 −1 −1 −1 x7

Figure II.2: A graph H with six vertices and the generalized Laplacian matrix of its cone.

Critical ideals of the complete graphs

We begin this subsection with an expression for the determinant of the complete graph.

Proposition 1.14. If Kn is the complete graph with n ≥ 1 vertices, then

det(L(Kn, X)) =

n∏j=1

(xj + 1)−n∑i=1

∏j 6=i

(xj + 1).

Proof. We will use induction on n. For n = 1, it is clear that det(L(Kn, X)) = x1 = (x1 + 1)− 1. Now,

assume that n ≥ 2. Expanding the determinant of L(Kn+1, X) by the last column and using induction

hypothesis

det(L(Kn+1, X)) = xn+1 · det(L(Kn, X)) +n∑k=1

det(L(Kn, X))xk=−1

= xn+1 ·

n∏j=1

(xj + 1)−n∑i=1

∏j 6=i

(xj + 1)

− n∑k=1

∏j 6=k

(xj + 1)

=n+1∏j=1

(xj + 1)−n∏j=1

(xj + 1)− (xn+1 + 1)n∑i=1

∏j 6=i

(xj + 1)

=

n+1∏j=1

(xj + 1)−n+1∑i=1

∏j 6=i

(xj + 1).

The next result gives us a description of a reduced Grobner basis of the critical ideals of the complete

graph.

Theorem 1.15. If Kn is the complete graph with n ≥ 2 vertices and 1 ≤ m ≤ n− 1, then

Bm ={∏i∈I

(xi + 1) | I ⊆ [n] and |I| = m− 1}

is a reduced Grobner basis of Im(Kn, X) with respect to the graded lexicographic order.


Proof. First, we will prove that Bm generates Im(Kn, X). If M is a square submatrix of L(Kn, X) of

size m, then there exist I ⊂ [n] with |I| = m and J ⊆ I such that M is equal to

L(Kn, X)[I; I]{xj=−1 for all j∈J}.

Thus, since

det(L(Kn, X))[I; I]{xj=−1 for all j∈J} =

∏ij∈I\J(xij + 1) if |J | = 1,

0 if |J | ≥ 2,

and det(L(Kn, X))[I; I] = −(xim+1)·det(L(Kn, X))[I; I]{xim=−1}+∑m

k=1 det(L(Kn, X))[I; I]{xik=−1},

then Bm generates Im(Kn, X).

Finally, we will prove that Bm is a reduced Grobner basis of Im(Kn, X) for all 1 ≤ m ≤ n − 1. Let

I1, I2 ⊂ [n] with |I1|, |I2| = m− 1 and pI =∏i∈I(xi + 1). It is not difficult to see that lt(pI) =

∏i∈I xi

and

pI1 ·∏

i∈I2\I1

(xi + 1)− pI2 ·∏

i∈I1\I2

(xi + 1) = 0.

Thus

S(pI1 , pI2) =lt(pI2)

lt(pI1∩I2)pI1 −

lt(pI1)

lt(pI1∩I2)pI2

=

∏i∈I2\I1

(xi + 1)−∏

i∈I2\I1

xi

· pI2 − ∏i∈I1\I2

(xi + 1)−∏

i∈I1\I2

xi

· pI1and S(pI1 , pI2)→Bm 0. Therefore, Bm is a reduced Grobner basis of Ii(Kn, X).

Remark. Note that, since∏∅ = 1, I1(Kn, X) = 〈1〉 and therefore γ(Kn) = 1.

Using the expression for the critical ideals of the complete graph given in Theorem 1.15, we can get

the primary decomposition of the critical ideals of the complete graph.

Corollary 1.16. If Kn is the complete graph with n ≥ 2 vertices and 1 ≤ m ≤ n− 1, then

Im(Kn, X) =⋂

I⊂[n],|I|=n−m+2

〈{xi + 1 | i ∈ I}〉.

As an application of Theorem 1.15, we find the critical group of all the graphs with ω(G) = |V (G)|−1.

Corollary 1.17. Let n > m ≥ 1 and Kn+1 \ Sm be the graph obtained from Kn+1 by deleting the m

edges of the star Sm. Then

K(Kn+1 \ Sm) =

Zn−2mn+1 ⊕ Zm−2

n(n+1) ⊕ Zn(n+1)(n−m) if m ≤ bn/2c ,

Z2m−nn ⊕ Zn−m−2

n(n+1) ⊕ Zn(n+1)(n−m) if m ≥ bn/2c .


Proof. Let vn+1 be the vertex of Kn+1 \ Sm of degree n−m. Thus

d = dKn+1−Sm(Kn+1 \ vn+1) = (n− 1, . . . , n− 1︸︷︷︸m times

, n, . . . , n︸︷︷︸n−m times

).

Now, by Proposition 1.14, In(Kn,d) = 〈nm−1(n+ 1)n−m−1(n−m)〉,

Ii+1(Kn,d) =

〈1〉 if i ≤ m,

〈(n+ 1)i−m〉 if m < i ≤ n−m,

〈ni+m−n(n+ 1)i−m〉 if n−m < i < n− 1,

when m ≤ bn/2c, and

Ii+1(Kn,d) =

〈1〉 if i ≤ n−m,

〈ni+m−n〉 if n−m < i ≤ m,

〈ni+m−n(n+ 1)i−m〉 if m < i < n− 1,

when m ≥ bn/2c. Finally, applying Proposition 1.7 we get the result.

To the authors knowledge, the critical group of these graphs had not been calculated before. We finish

this section by exploring a relation between the critical ideals and the characteristic polynomial of the

adjacency matrix of a graph.

Critical ideals and the characteristic polynomials

In [32], Lorenzini showed a deep relation between the critical group and the Laplacian spectrum of a

graph. For instance, Lorenzini ([32, Proposition 3.2]) proved that if λ is an integer eigenvalue of L(G)

of multiplicity µ(λ), then K(G) contains a subgroup isomorphic to Zµ(λ)−1λ . In this subsection, we

will present the relation that exists between the critical ideals and the characteristic polynomial of the

adjacency matrix of a graph. If G is loopless and we take xi = t for all 1 ≤ i ≤ n, then det(L(G,X)) is

equal to the characteristic polynomial pG(t) of the adjacency matrix of G and the critical ideals of G

are ideals in P[t]. Let Ii(G, t) = Ii(G,X){xj=t | ∀ 1≤j≤n} for all 1 ≤ i ≤ n. It is not difficult to see that

if P is a field, then the ideals Ii(G, t) are principal. That is, there exist pi(t) ∈ P[t] for all 1 ≤ i ≤ n

such that

Ii(G, t) = 〈i∏

j=1

pj(t)〉 for all 1 ≤ i ≤ n.

Thus, pG(t) =∏nj=1 pj(t) is a factorization of the characteristic polynomial of the adjacency matrix of

G. In a similar way, if we take xi = di − t for all 1 ≤ i ≤ n, then we recover a factorization of the

characteristic polynomial of the Laplacian matrix of G from their critical ideals.

Therefore, the critical ideals of a graph are a generalization of the characteristic polynomial the adja-

cency matrix and the Laplacian matrix of G. For instance, if G is the complete graph with six vertices

Section 2 : Critical ideals of the cycle 35

minus a perfect matching (as in Example 1.2) and P = Q, then

Ii(G, t) =

〈1〉 if 1 ≤ i ≤ 4,

〈t2(t+ 2)〉 if i = 5,

〈t3(t+ 2)2(t− 4)〉 if i = 6.

Note that in general the t-critical ideals depend on the base ring. For instance, if P = Z, then

Ii(G, t) =

〈1〉 if i = 1, 2,

〈2, t〉 if i = 3,

〈t2, 4t〉 if i = 4,

〈t3(t+ 2), 4t2(t+ 2)〉 if i = 5,

〈t3(t+ 2)2(t− 4)〉 if i = 6.

The critical ideals are a stronger invariant than the adjacency spectrum of the graph. For instance, if

v1 v2

v3

v4

v5 v6G1

v1 v2

v3

v4

v5 v6G2

Figure II.3: The graphs G1 and G2 have the same characteristic polynomial.

P = Z, G1 and G2 are the graphs given in Figure II.3, then they are cospectral, but

Ii(G1, t) =

〈1〉 if 1 ≤ i ≤ 4,

〈2(t+ 1), (t+ 1) · (t2 + 1)〉 if i = 5,

〈(t− 1) · (t+ 1)2 · (t3 − t2 − 5t+ 1)〉 if i = 6,

and

Ii(G2, t) =

〈1〉 if 1 ≤ i ≤ 3,

〈2, (t+ 1)〉 if i = 4,

〈4(t+ 1), (t+ 1) · (t− 3)〉 if i = 5,

〈(t− 1) · (t+ 1)2 · (t3 − t2 − 5t+ 1)〉 if i = 6.

2. Critical ideals of the cycle

The main goal of this section is to get a minimal set of generators and the reduced Grobner basis of

the critical ideals of the cycle. Before of doing this, we will get some combinatorial expressions for the

minors of the generalized Laplacian matrix of a digraph. These expressions will be very useful in order

to get some algebraic relations between the generators of the critical ideals of the cycle.


We begin with some concepts of digraphs. Given a digraph D, a subdigraph C of D is called a directed

1-factor if and only if d+C(v) = d−C(v) = 1 for all v ∈ V (C). The number of connected components of

C will be denoted by c(C).

Theorem 2.1. If D is a digraph with n vertices (possibly with multiple arcs and loops), then

det(−A(D)) =∑C∈→F

(−1)c(C),

where→F is the set of spanning directed 1-factors of D. Moreover,

det(L(D,X)) =∑

U⊆V (D)

det(−A(D[U ]wl)) ·∏v/∈U

xv,

where D[U ] is the induced subgraph of D by U and D[U ]wl is the graph obtained from D[U ] when we

delete the possible loops.

Proof. It follows from arguments similar to those in [10, Proposition 7.2].

Remark. If we identify a graph G with the digraph DG, obtained from G by replacing each edge uv of

G by the arcs−→uv and

−→vu, then an elementary graph is identified with a spanning directed 1-factor and

therefore the first part of Theorem 2.1 is equivalent to [10, Proposition 7.2]. In this way, Theorem 2.1

generalizes the expression obtained in [10, Proposition 7.2].

Next we will present an example of the use of Theorem 2.1.

Example 2.2. If D is the digraph given by V (D) = {v1, v2, v3, v4} (see Figure II.4) and

E(D) = { −→v1v1,−→v1v2,

−→v2v3,

−→v3v4,

−→v4v1,

−→v4v2},

then D has two spanning directed 1-factors; { −→v1v2,−→v2v3,

−→v3v4,

−→v4v1} and { −→v1v1,

−→v2v3,

−→v3v4,

−→v4v2}. Thus,

det(A(D)) = (−1)1 + (−1)2 = 0. Also, since { −→v1v2,−→v2v3,

−→v3v4,

−→v4v1} and { −→v2v3,

−→v3v4,

−→v4v2} are the

directed 1-factors of Dwl, det(L(D,X)) = x1x2x3x4 − x1 − 1.

v1

v2 v3

v4

Figure II.4: A digraph with four vertices.

When G is a tree, the determinant of the generalized Laplacian matrix given in Theorem 2.1, only

depends on its matchings. Therefore, in this case we can get an explicit combinatorial expression for

the determinant of the generalized Laplacian. Given a set of edges µ, let V (µ) be the set of vertices of

the induced subgraph by µ.


Lemma 2.3. If T is a tree, then

det(L(T,X)) =∑

µ∈V1(T )

(−1)|µ|∏

v/∈V (µ)

xv,

where V1(T ) is the set of matchings on T .

Proof. It follows from Theorem 2.1 and the fact that if S is a forest, then

det(A(S)) =

(−1)|µ| if µ is a perfect matching of S, and

0 otherwise.

Moreover, Theorem 2.1 and Lemma 2.3 can be used to get combinatorial expressions for the determi-

nants of the generalized Laplacian matrices of graphs.

Corollary 2.4. Let n be a natural number, Kn be the complete graph with n vertices, Pn be the path

with n vertices, and Cn the cycle with n vertices. Then

(i) det(L(Kn, X)) =∑I⊆[n]

(|I| − n+ 1) ·∏i∈I

xi,

(ii) det(L(Pn, X)) =∑

µ∈V1(Pn)

(−1)|µ| ·∏

v 6∈V (µ)

xv,

(iii) det(L(Cn, X)) =∑

µ∈V1(Cn)

(−1)|µ| ·∏

v 6∈V (µ)

xv − 2,

where V1(G) is the set of matchings of G.

Proof. (i) Follows by Theorem 2.1 and the fact that det(A(Km)) = −m+ 1 for all m ≥ 2. (ii) Follows

directly from Lemma 2.3. (iii) Let P li be the induced path of Cn of length l that begins in the vertex i

of Cn. Since V1(Cn) = V1(Cn \ v1vn) t {µ ∈ V1(Cn) | v1vn ∈ µ},

det(L(Cn, X)) = det(L(Pn1 , X))− det(L(Pn−22 , X))− 2 =

∑µ∈V1(Cn)

(−1)|µ| ·∏

v 6∈V (µ)

xv − 2.

We will show that every minor of a generalized Laplacian of a digraph is equal to an evaluation of the

determinant of the generalized Laplacian of some digraph.

Given a digraph (or a graph) D and u 6= v two vertices of D, let D(u; v) be the digraph obtained from

D by deleting the arcs leaving u and entering v (remember that each edge of a graph is considered as

two arcs in both directions) and identifying the vertices u and v in a new vertex, denoted by u ◦ v.

Also, if u = v, then D(u; v) is defined as D \ u.

Example 2.5. If Pn is the path with n vertices (simple graph), then

Pn(v1; vn) ∼= Cn−1 \ {−→v1v2,

−→vn−1v1}.


Now, given two matrices M,N ∈ Mm×m(P), we say that M and N are strongly equivalent, denoted

by N ≈ M , if there exist P and Q permutation matrices such that N = PMQ. Also, given U =

{u1, . . . , us} and V = {v1, . . . , vs} ordered subsets of V (D) with s ≥ 2 , we define D(U ;V ) inductively

as D(U \ us;V \ vs)(us; vs).

Lemma 2.6. If D is a digraph and U = {u1, . . . , us}, V = {v1, . . . , vs} ordered subsets of V (D), then

L(D,X)(U ;V ) ≈ L(D(U ;V ), X){xui◦vi=−m(vi,ui)| i=1,...,s}.

Proof. It follows from the construction of D(U ;V ).

Remark. Clearly, D(U ;V ) and L(D(U ;V ), X) depend on the order of the elements of the subsets U

and V . However, if σ ∈ Ss is a permutation, then

L(D(U ;V ), X){xui◦vi=−m(vi,ui)| i=1,...,s} ≈ L(D(U ; {vσ(1), . . . , vσ(s)}), X){xui◦vσ(i)=−m(vσ(i),ui)

| i=1,...,s},

that is, in some sense L(D(U ;V ), X){xui◦vi=−m(vi,ui)| i=1,...,s} does not depend on the order of the

elements of the subsets U and V .

Example 2.7. Let D be the cycle with six vertices, U = {v1, v2}, and V = {v6, v5}. Then we get

that D(U ;V ) is the digraph with four vertices {v3, v4, v2 ◦ v5, v1 ◦ v6} (see Figure II.5(a)) and arcs

{v3v4, v4v3, v3(v2 ◦ v5), (v1 ◦ v6)v4}. On the other hand, if we change the order of the elements in U ,

then D(U ;V ) is given by the graph in Figure II.5(b).

v1 ◦ v6v2 ◦ v5

v3

v4v1 ◦ v5v2 ◦ v6

v3 v4

(a) (b)

Figure II.5: Digraphs with four vertices.It is not difficult to see that

L(D,X)(U ;V ) =

0 −1 x3 −1

0 0 −1 x4

0 0 0 −1

−1 0 0 0

≈

x3 −1 −1 0

−1 x4 0 0

0 −1 0 0

0 0 0 −1

= L(D({v1, v2}; {v6, v5}), X){xv2◦v5=0, xv1◦v6=−1}.

≈

x3 −1 −1 0

−1 x4 0 0

0 0 0 −1

0 −1 0 0

= L(D({v2, v1}; {v6, v5}), X){xv2◦v6=0, xv1◦v5=0}.

In a similar way, we can define the digraph D[U ;V ] that satisfies that

L(D,X)[U ;V ] ≈ L(D[U ;V ], X){xui◦vi=−m(vi,ui)| i=1,...,s}.

In the next results, we calculate the invariant γ for the path using the previous results on the minors

of a generalized Laplacian matrix.


Corollary 2.8. Let Pn be the path with n vertices, then γP(Pn) = n− 1.

Proof. It follows because det(L(Pn, X)(1;n)) = 1.

Corollary 2.9. Let G be a simple graph with Pn as an induced graph, then γP(G) ≥ n− 1.

Proof. It follows directly from Proposition 1.3 and Corollary 2.8.

A crucial open question discussed in [32, Section 4] and [37, Section 4] states that the critical group

is cyclic for almost all simple graphs. However, after computing the critical ideals of all simple graphs

with less or equal to 9 vertices, we conjecture that the only simple graph with γP(G) = n − 1 is the

path with n vertices.

Conjecture 2.10. Let G be a simple graph with n vertices and Z ⊆ P, then γP(G) = n − 1 if and

only if G = Pn.

Critical ideals of the cycle

In this subsection, we will calculate the critical ideals of the cycle with n vertices. Let Cn be the

cycle with n vertices and let V (C) = {v1, v2, . . . , vn} be its vertex set. To simplify the notation, we

consider the vertices of Cn as the classes modulo n. That is, for instance, the (n+ 1)-th vertex of Cn

is the vertex v1. Clearly, In(Cn, X) is generated by the determinant of L(Cn, X). In Corollary 2.4,

the determinant of the generalized Laplacian of Cn was calculated. Therefore, the critical ideals of Cn,

when 1 ≤ i ≤ n− 1, are the only ones that remain to calculate. In order to simplify the notation, we

will write det(G,X) instead of det(L(G,X)).

First we will prove that almost all the critical ideals of Cn are trivial, except for i equal to n− 1 and n.

Also, we will give a minimal set of generators for In−1(Cn, X), and after that, we will give a reduced

Grobner basis for it.

Theorem 2.11. If Cn is the cycle with n vertices, then Ii(Cn, X) = 〈1〉 for all 1 ≤ i ≤ n−2. Moreover,

Fk = {det(Cn\{vk+1, vk, vk−1}, X)+xk,det(Cn\{vk, vk+1vk+2}, X)+xk+1,det(Cn\{vk, vk+1}, X)+1}

for all 1 ≤ k ≤ n is a minimal set of generators for In−1(Cn, X).

Proof. Firstly, we will prove that Ii(Cn, X) = 〈1〉 for all 1 ≤ i ≤ n− 2. Let D be the digraph obtained

from the cycle Cn−2 with n− 2 vertices, when we delete the arcs−→v1v2 and

−→vn−2v1. It is not difficult to

see that

L(Cn, X)({1, n}; {n− 1, n}) ≈ L(D,X){x1=0}.

By applying Theorem 2.1 to the digraph D, we have that |det(L(Cn, X)({1, n}; {n − 1, n}))| = 1 and

therefore Ii(Cn, X) = 〈1〉 for all 1 ≤ i ≤ n− 2.

Now, we turn our attention to the (n − 1)-th critical ideal of Cn. For all 1 ≤ i, j ≤ n, let Qi,j =

L(Cn, X)(i; j) and

qi,j =

−det(Qi,j) if lc(det(Qi,j)) = −1,

det(Qi,j) otherwise.


Since L(Cn, X) is symmetric, Qi,j ≈ Qj,i and qi,j = qj,i. Clearly, the ideal In−1(Cn, X) is generated by

the n2 minors of size n− 1 of L(Cn, X).

On the other hand, for all 1 ≤ i ≤ n and 0 ≤ s ≤ n− 1, let P si be the induced path with s vertices of

Cn consisting of the vertices vi, vi+1, . . . , vi+s−1. For instance, in C6 we have that P 45 consists of the

vertices v5, v6, v1, and v2. Note that, if s = 0, then P si is the empty graph.

Definition 2.12. For all 1 ≤ i ≤ n and 2 ≤ s ≤ n, let pi,s = det(P s−1i , X). For technical convenience

we will adopt the convention that pi,1 = 1 for all 1 ≤ i ≤ n and pi,s = −pi+s,−s for all s ≤ 0.

Note that, by taking s = 0 we get that pi,−0 = −pi,0 and therefore pi,0 = 0 for all 1 ≤ i ≤ n. Also,

taking s = −1, we get that pi,−1 = −pi−1,1 = −1. Finally, note that lt(pi,s) = xixi+1 · · ·xi+s−2 for all

s ≥ 2 and deg(pi,s) = s−1 for all s ≥ 1. The rest of this chapter relies heavily in several identities that

involved the qi,j ’s and the pi,s’s. In order to do more understandable the meaning of these identities

it will be very helpful to think the polynomial qi,j as a polynomial associated to the edge vivj (see

Figure II.6). The first one of these identities is the following:

Claim 2.13. If 1 ≤ i ≤ j ≤ n, then

qi,j = pi+1,j−i + pj+1,n−j+i.

Proof. At first, if j = i, then clearly Qi,i = L(Cn, X)(i; i) = L(Cn \ vi, X \ xi) = L(Pn−1i+1 , X). Thus

qi,i = pi+1,n = pi+1,0 + pi+1,n. Now, if j = i+ 1, then by Lemma 2.6

Qi,i+1 = L(Cn, X)(i; i+ 1) ≈ L(D,X){xvi◦vi+1=−1},

where D is the digraph obtained from Cn by deleting the arcs−→

vivi−1,−→

vi+2vi+1 and identifying the

vertices i and i+1. Since D has only one spanning directed 1-factor (a directed cycle) and all the other

directed 1-factors of D are 1-factors of D \ vi ◦ vi+1∼= Pn−2, then by applying Theorem 2.1 to D we

get that det(L(D,X)){xvi◦vi+1=−1} = −pi+2,n−1 − 1, and therefore qi,i+1 = pi+1,1 + pi+2,n−1. Finally, if

j 6= i, i+ 1, then by Lemma 2.6

L(Cn, X)(i; j) ≈ L(D,X){xvi◦vj=0},

where D is the digraph obtained from Cn by deleting the arcs−→

vivi+1,−→

vj−1vj ,−→

vivi−1,−→

vj+1vj and identi-

fying the vertices i and j. Since D has only two directed cycles containing the vertex vi ◦ vj , then by

applying Theorem 2.1 we get the result.

From Claim 2.13 can be interpreted that qi,j is equal to the sum of the determinants of the generalized

Laplacian matrices of the two paths obtained when we delete the vertices vi and vj to Cn. The next

algebraic identity will be key for the rest of the chapter. For instance, it will be useful to find a minimal

set of generators for In−1(Cn, X).

Claim 2.14. If 1 ≤ i ≤ n, 2 ≤ j ≤ n− 1, and −j ≤ s ≤ n− j, then

qi,i+j+s = pi+j,s+1 · qi,i+j − pi+j+1,s · qi,i+j−1. (II.1)

That is, 〈{qi,t}nt=1〉 = 〈qi,i+j , qi,i+j−1〉.


Proof. Note that by Definition 2.12, the Equation II.1 we can be written as qi,i+j+s = −pi+j+s+1,−s−1 ·qi,i+j + pi+j+s+1,−s · qi,i+j−1 for all s ≤ −1. The result is trivial for s = 0 and s = −1. We divide the

proof in two cases: when s ≥ 0 and s ≤ −1. For both cases we will use induction on s.

v1

vi

vi+j+s (s ≤ −1)

vi+j−1qi,i+j−1vi+j

qi,i+j

pi+j,s+1 pi+j+1,s

vi+j+s

vn

qi,i+j+s

pi+j+s+1,−s

pi+j+s+1,−s−1

Cn

Figure II.6: A cycle Cn with n vertices illustrating the identity qi,i+j+s = −pi+j+s+1,−s−1 · qi,i+j +

pi+j+s+1,−s · qi,i+j−1, on the qi,j ’s and pi,s’s polynomials given in claim 2.14.

We begin with the case when s ≥ 0. If s = 1, then expanding the determinant of L(P ji+1, X) and

L(Pn−ji+j , X) we have that pi+1,j+1 = xi+j · pi+1,j − pi+1,j−1 and pi+j,n−j+1 = xi+j · pi+j+1,n−j −pi+j+2,n−j−1. Therefore

xi+jqi,i+j2.13= xi+j · [pi+1,j + pi+j+1,n−j ] = pi+1,j−1 + pi+1,j+1 + pi+j,n−j+1 + pi+j+2,n−j−1

= pi+1,j−1 + pi+j,n−j+1 + pi+1,j+1 + pi+j+2,n−j−1 = qi,i+j−1 + qi,i+j+1.

Now, assume that s ≥ 2. Using induction hypothesis we get

qi,i+j+s+1 = xi+j+s · qi,i+j+s − qi,i+j+s−1

= xi+j+s · (pi+j,s+1 · qi,i+j − pi+j+1,s · qi,i+j−1)− (pi+j,s · qi,i+j − pi+j+1,s−1 · qi,i+j−1)

= (xi+j+s · pi+j,s+1 − pi+j,s) · qi,i+j − (xi+j+s · pi+j+1,s − pi+j+1,s−1) · qi,i+j−1

= pi+j,s+2 · qi,i+j − pi+j+1,s+1 · qi,i+j−1.

Similar arguments can be used when s ≤ −1.

Note that Claim 2.14 gives us two different expression for qi,i. Namely, qi,i = pi+j,n−j+1 · qi,i+j −pi+j+1,n−j · qi,i+j−1 = pi+1,j · qi,i+j−1 − pi+1,j−1 · qi.i+j .Thinking the qi,j ’s polynomials as a polynomial associated to the edge vivj , we have that Claim 2.14

says us that the polynomial associated to any edge adjacent to vi can be expressed as a polynomial

combination of the two polynomial associated to two consecutive edges adjacent to vi. Thus, if we fix

i and j (with i 6= j, j + 1), then for all 1 ≤ t ≤ n, qi,t can be generated by qi,j and qi,j+1. That is,

〈{qi,t}nt=1〉 = 〈qi,j , qi,j+1〉 for all 1 ≤ i, j ≤ n with i 6= j, j + 1. Therefore, it is not difficult to see that

In−1(Cn, X) = 〈qi,i+1, qi,i+2, qi−1,i+1〉 for all 1 ≤ i ≤ n.

Next example presents how claim 2.14 can be used.

Example 2.15. For the cycle with 6 vertices we have that


1. q1,1 = x2x3x4x5x6 − x2x3x4 − x2x3x6 −x2x5x6 − x4x5x6 + x2 + x4 + x6,

2. q1,2 = x3x4x5x6 − x3x4 − x3x6 − x5x6 + 2,

3. q1,3 = x4x5x6 + x2 − x4 − x6,

4. q1,4 = x2x3 + x5x6 − 2,

5. q1,5 = x2x3x4 − x2 − x4 + x6,

6. q1,6 = x2x3x4x5 − x2x3 − x2x5 − x4x5 + 2.

It is not difficult to see that

• q1,1 = p4,−2 · q1,4 − p5,−3 · q1,3 = −p2,2 · q1,4 + p2,3 · q1,3 = −x2 · q1,4 + (x2x3 − 1) · q1,3,

• q1,2 = p4,−1 · q1,4 − p5,−2 · q1,3 = −p3,1 · q1,4 + p3,2 · q1,3 = −q1,4 + x3 · q1,3,

• q1,3 = p4,0 · q1,4 − p5,−1 · q1,3,

• q1,4 = p4,1 · q1,4 − p5,0 · q1,3,

• q1,5 = p4,2 · q1,4 − p5,1 · q1,3 = x4 · q1,4 − q1,3,

• q1,6 = p4,3 · q1,4 − p5,2 · q1,3 = (x4x5 − 1) · q1,4 − x5 · q1,3,

• q1,1 = p4,4 · q1,4 − p5,3 · q1,3 = (x4x5x6 − x4 − x6) · q1,4 − (x5x6 − 1) · q1,3.

Before to present a Grobner basis for In−1(Cn, X), we present an identity between the qi,j ’s. These

identities will be useful to reduce the S-polynomials of the qi,j ’s.

Proposition 2.16. Let n ≥ 6, 0 ≤ i, s ≤ n, and 2 ≤ j, t ≤ n− 2.

(i) If 0 ≤ s ≤ j − 1, then (see Figure II.7 (i))

pi+s+1,j−s ·qi+j−1,i−pi+s+t,n−s−t+1 ·qi+s,i+s+t = pi+s+1,j−s−1 ·qi+j,i−pi+s+t+1,n−s−t ·qi+s,i+s+t−1.

(ii) If j ≤ s ≤ n, then (see Figure II.7 (ii))

pi+j,s−j+1 · qi,i+j − pi+1,s+t−n · qi+s+t−1,i+s = pi+j+1,s−j · qi,i+j−1 − pi+1,s+t−n−1 · qi+s+t,i+s.

Proof. (i) By using Claim 2.14 when 0 ≤ s ≤ j − 1, we have that

qi,i+s = pi+s+1,j−s · qi,i+j−1 − pi+s+1,j−s−1 · qi,i+j

and

qi+s,i = pi+s+t,n−s−t+1 · qi+s,i+s+t − pi+s+t+1,n−s−t · qi+s,i+s+t−1.

By using the fact that qi,j = qj,i in the first identity, we get that

pi+s+1,j−s · qi+j−1,i − pi+s+t,n−s−t+1 · qi+s,i+s+t = pi+s+1,j−s−1 · qi+j,i − pi+s+t+1,n−s−t · qi+s,i+s+t−1.


v1vn

vi

vi+s

vi+s+t

vi+s+t−1

vi+j−1

vi+j

pi+s+1,j−s

pi+s+1,j−s−1

pi+s+t,n−s−t+1

pi+s+t+1,n−s−t

qi,i+s

v1

vi

vi+s+t−1

vi+s+tvi+j−1

qi,i+j−1

qi+s,i+s+t

vi+j

pi+j,s−j+1 pi+1,s+t−nqi,i+j

vi+s

vn

qi,i+s

(i) 0 ≤ s ≤ j − 1, (ii) j ≤ s ≤ n.

Figure II.7: A cycle with n vertices illustrating the identity given in Proposition 2.16.

(ii) In a similar way, if j ≤ s ≤ n, then by Claim 2.14

qi,i+s = pi+j,s−j+1 · qi,i+j − pi+j+1,s−j · qi,i+j−1

and

qi+s,i = pi+1,s+t−n · qi+s,i+s+t−1 − pi+1,s+t−n−1 · qi+s,i+s+t

By using the fact that qi,j = qj,i in the second identity we get that

pi+j,s−j+1 · qi,i+j − pi+1,s+t−n · qi+s+t−1,i+s = pi+j+1,s−j · qi,i+j−1 − pi+1,s+t−n−1 · qi+s+t,i+s.

Remark. Note that the two identities in Proposition 2.16 are equivalents in the sense that only differ

by a rotation on the labels of the vertices.

Now, we are ready to find a Grobner basis for In−1(Cn, X). In general, Grobner basis is very useful,

see for instance [22]. For instance, almost all the information about an ideal can be extracted from its

Grobner basis. We divide this into odd and even cases.

Theorem 2.17. Let n = 2m+ 1 ≥ 7 and

B1 = {qi,i+m+1 | i = 1, . . . , n} = {qi,i+m | i = 1, . . . , n}.

Then B1 is a reduced Grobner basis for In−1(Cn, X) with respect to any graded lexicographic order.

Proof. First, since qi,i+m and qi,i+m+1 are in B1 for all 1 ≤ i ≤ n, then it is not difficult to see that

In−1(Cn, X) = 〈B1〉. On the other hand, by Claim 2.13, lt(qi,i+m+1) = lt(pi+1,m+1) = xi+1 · · ·xi+m.

Thus deg(qi,i+m+1) = m and lt(qi,i+m+1) - lt(qi′,i′+m+1) for all 1 ≤ i < i′ ≤ n. Moreover, deg(qi,i+m+1−lt(qi,i+m+1)) = m− 1 for all 1 ≤ i ≤ n. Therefore, if B1 is a Grobner basis for In−1(Cn, X), then it is

reduced.


In order to prove that B1 is a Grobner basis for In−1(Cn, X), we need to show that all the S-polynomialsof the elements on B1 can be reduced to 0 by elements on B1. Let 1 ≤ i ≤ n and 1 ≤ r ≤ n− 1, it isnot difficult to see (Figure II.8) that

S(qi,i+m+1, qi+r,i+m+r+1) =

(xi+m+1 · · ·xi+m+r) · qi,i+m+1 − (xi+1 · · ·xi+r) · qi+r,i+m+r+1 if 1 ≤ r ≤ m,

(xi+r+1 · · ·xi+n) · qi,i+m+1 − (xi+r+m+1 · · ·xi+m) · qi+r,i+m+r+1 if m+ 1 ≤ r ≤ 2m,

=

lt(pi+m+1,r+1) · qi,i+m+1 − lt(pi+1,r+1) · qi+r,i+m+r+1 if 1 ≤ r ≤ m,

lt(pi+r+1,n−r+1) · qi,i+m+1 − lt(pi+r+m+1,n−r+1) · qi+r,i+m+r+1 if m+ 1 ≤ r ≤ 2m.

By Proposition 2.16 with j = t = m+ 1 and s = m+ r + 1 in the second identity and j = t = m+ 1,

v1

vi

vi′vj

lt(pj,j′−j+1) lt(pi+1,i′−i+1)

qi,j qi′,j′gcd(lt(qi,j), lt(qi′,j′ ))

vj′

vn

Figure II.8: The S-polynomial of qi,j and qi′,j′ .

s = i+m+ r + 1 in the first identity we get that

S(qi,i+m+1, qi+r,i+m+r+1) =

−red(pi+m+1,r+1) · qi,i+m+1 + red(pi+1,r+1) · qi+r,i+m+r+1 + pi+m+2,r · qi,i+m−pi+1,r · qi+r+1,i+m+r+1, if 1 ≤ r ≤ m,

−red(pi+r+1,n−r+1) · qi,i+m+1 + red(pi+r+m+1,n−r+1) · qi+r,i+m+r+1

+pi+r+1,n−r · qi+1,i+m+1 − pi+m+r+2,n−r · qi+r,i+m+r, if m+ 1 ≤ r ≤ 2m,

where red(p) = p− lt(p) for any polynomial p.

Since qi,i+m+1, qi+r,i+m+r+1, qi,i+m, qi+r+1,i+m+r+1, qi+1,i+m+1, qi+r,i+m+r ∈ B1 for all 1 ≤ i ≤ n and

1 ≤ r ≤ n−1, then in order to prove that S(qi,i+m+1, qi+r,i+m+r+1)→B1 0, it only remains to prove that

the leading terms of the summands of previous identity are different. Since lt(pi+m+1,r+1 · qi,i+m+1) =

lt(pi+1,r+1 ·qi+r,i+m+r+1), lt(pi+r+1,n−r+1 ·qi,i+m+1) = lt(pi+r+m+1,n−r+1 ·qi+r,i+m+r+1) are square free,

gcd(lt(pi+m+1,r+1), lt(pi+1,r+1)) = 1, gcd(lt(pi+r+1,n−r+1), lt(pi+r+m+1,n−r+1)) = 1,

red(pi+1,r+1) = xi+1 · · ·xi+r/xjxj+1 for some i+ 1 ≤ j ≤ i+ r − 1,

and

1. lt(pi+m+2,r · qi,i+m) = xi+m+1x2i+m+2 · · ·x2

i+m+rxi+m+r+1 · · ·xi−1,

2. lt(pi+1,r · qi+r+1,i+m+r+1) = xi+m+r+2 · · ·xix2i+1 · · ·x2

i+r−1xi+r,

3. lt(pi+r+1,n−r · qi+1,i+m+1) = xi+m+2 · · ·xi+rx2i+r+1 · · ·x2

i−1xi,

4. lt(pi+m+r+2,n−r · qi+r,i+m+r) = xi+m+r+1x2i+m+r+2 · · ·x2

i+mxi+m+1 · · ·xi+r−1,

then S(qi,i+m+1, qi+r,i+r+m+1)→B1 0 for all 1 ≤ i ≤ n− 1 and 1 ≤ r ≤ n− i.


If n = 3, then Cn is the complete graph with three vertices. Now we will present the case of the cycle

with five vertices.

Example 2.18. Let n = 5 = 2(2) + 1 = 2m + 1 and B1 as in Theorem 2.17. It is not difficult to

compute that

1. q1,4 = x2x3 − 1 + x5,

2. q2,5 = x3x4 − 1 + x1,

3. q3,1 = x4x5 − 1 + x2,

4. q4,2 = x5x1 − 1 + x3,

5. q5,3 = x1x2 − 1 + x4.

Moreover, since

• S(q1,4, q2,5) = x4 · q1,4 − x2 · q2,5 = q3,1 − q5,3 →B1 0,

• S(q1,4, q3,1) = x4x5 · q1,4 − x2x3 · q3,1 = (x5 − 1) · q3,1 − (x2 − 1) · q1,4 →B1 0,

• S(q1,4, q4,2) = x5x1 · q1,4 − x2x3 · q4,2 = (x5 − 1) · q4,2 − (x3 − 1) · q1,4 →B1 0,

• S(q1,4, q5,3) = x1 · q1,4 − x3 · q5,3 = q4,2 − q2,5 →B1 0,

• S(q2,5, q3,1) = x5 · q2,5 − x3 · q3,1 = q4,2 − q1,4 →B1 0,

• S(q2,5, q4,2) = x5x1 · q2,5 − x3x4 · q4,2 = (x1 − 1) · q4,2 − (x3 − 1) · q2,5 →B1 0,

• S(q2,5, q5,3) = x1x2 · q2,5 − x3x4 · q5,3 = (x1 − 1) · q5,3 − (x4 − 1) · q2,5 →B1 0,

• S(q3,1, q4,2) = x1 · q3,1 − x4 · q4,2 = q5,3 − q2,5 →B1 0,

• S(q3,1, q5,3) = x1x2 · q3,1 − x4x5 · q5,3 = (x2 − 1) · q5,3 − (x4 − 1) · q3,1 →B1 0,

• S(q4,2, q5,3) = x2 · q4,2 − x5 · q5,3 = q1,4 − q3,1 →B1 0,

then B1 is a Grobner basis for I4(C5).

The even case is slightly different.

Theorem 2.19. Let n = 2m ≥ 6 and

B0 = {qi,i+m | i = 0, . . . ,m− 1} ∪ {qi+m,i+1 | i = 0, . . . ,m− 2}.

Then B0 is a Grobner basis for In−1(Cn, X) with respect to the graded lexicographic order with xm−1 >

· · · > x2m > x1 · · · > xm−2.

Proof. We have that qi,i+m, qi,i+m−1 ∈ B0 for all i = 1, · · · ,m − 1 and qi,i−m, qi,i−m+1 ∈ B0 for all

i = m, · · · , 2m − 2. Then by Claim 2.14, {qi,j}nj=1 ( 〈B0〉 for all i = 1, · · · , 2m − 2. Moreover,

since q2m,1, q2m,2 ∈ 〈B0〉 and q2m−1,1, q2m−1,2 ∈ 〈B0〉, then by Claim 2.14, {qi,j}nj=1 ( 〈B0〉 for i =

2m − 1, 2m and therefore In−1(Cn, X) = 〈B0〉. Following the proof of Theorem 2.17, we have that

lt(qi,i+m) = lt(pi+1,m) = xi+1 · · ·xi+m−1 for all i = 0, . . . ,m − 1 and lt(qi+m,i+1) = lt(pi+m+1,m+1) =

xi+m+1 · · ·xi for all i = 0, . . . ,m − 2. Thus lt(qi,i+m) - lt(qi′,i′+m) for all 0 ≤ i < i′ ≤ m − 1,


lt(qi+m,i+1) - lt(qi′+m,i′+1) for all 0 ≤ i < i′ ≤ m− 2, and lt(qi,i+m) - lt(qi′+m,i′+1) for all 0 ≤ i ≤ m− 1

and 0 ≤ i′ ≤ m − 2. Moreover, since deg(red(qi,i+m)) = m − 1, deg(red(qi+m,i+1)) = m − 2 and

lt(red(qi,i+m)) = lt(pi+m+1,m) = xi+m+1 · · ·xi−1, when B0 is a Grobner basis for In−1(Cn, X), then it

is reduced.

In a similar way to the proof of Theorem 2.17, we need to show that all the S-polynomials of the

elements on B0 can be reduced to 0 by elements on B0. At difference to the proof of Theorem 2.17, in

this case we have three types of S-polynomials of elements of B0. First

S(qi,i+m, qi+s,i+m+s) = (xi+m · · ·xi+m+s−1) · qi,i+m − (xi+1 · · ·xi+s) · qi+s,i+m+s

= lt(pi+m,s+1) · qi,i+m − lt(pi+1,s+1) · qi+s,i+m+s

for all 0 ≤ i ≤ m−2 and 1 ≤ s ≤ m− i−1. Applying Proposition 2.16 (ii) with j, t = m and s = m+s

we get that

pi+m,s+1 · qi,i+m − pi+1,s+1 · qi+s,i+m+s = pi+m+1,s · qi,i+m−1 − pi+1,s · qi+s+1,i+m+s.

Thus

S(qi,i+m, qi+s,i+m+s) = −red(pi+m,s+1) · qi,i+m + red(pi+1,s+1) · qi+s,i+m+s

+pi+m+1,s · qi,i+m−1 − pi+1,s · qi+s+1,i+m+s. (II.2)

Since qi,j = qj,i, then it is not difficult to use Equation II.2 to prove that S(qi,i+m, qi+s,i+m+s) is reduced

to 0 by B0 whenever i ≥ 1 and i+ s ≤ m− 2. Now, it only remains to analyze two special cases, when

i = 0 and i+ s = m− 1. Taking i = 0 in Equation II.2 and the fact that

q0,m−1 = qm−1,2m2.14= p2m−1,2 · qm−1,2m−1 − p2m,1qm−1,2m−2 = x2m−1 · qm−1,2m−1 − qm−1,2m−2

we have that

S(q0,m, qs,m+s) = −red(pm,s+1) · q0,m + red(p1,s+1) · qs,m+s + x2m−1 · pm+1,s · qm−1,2m−1

−

pm+1,s · qm−1,2m−2 + p1,s · qs+1,m+s if s ≤ m− 3,

(pm+1,m−2 + p1,m−2) · qm−1,2m−2 if s = m− 2.

Therefore, S(q0,m, qs,m+s) →B0 0 for all 1 ≤ s ≤ m − 2. On the other hand, taking i + s = m − 1 in

Equation II.2 and using that qm,2m−1 = q2m−1,m2.14= x2m · q0,m − q1,m, we have that

S(qi,i+m, qm−1,2m−1) = −red(pi+m,s+1) · qi,i+m + red(pi+1,s+1) · qm−1,2m−1

+pi+m+1,s · qi,i+m−1 − pi+1,s · qm,2m−1

= −red(pi+m,s+1) · qi,i+m + red(pi+1,s+1) · qm−1,2m−1

+

(pm+2,s + p2,s) · q1,m − x2m · p2,s · q0,m if i = 1,

pi+m+1,s · qi,i+m−1 − x2m · pi+1,s · q0,m + pi+1,s · q1,m if i ≥ 2.

Therefore, S(qi,i+m, qm−1,2m−1) can be reduced to 0 by B0. Finally, if i = 0 and s = m− 1, then

S(q0,m, qm−1,2m−1) = red(p0,m) · qm−1,2m−1 − red(pm−1,2m−1) · q0,m.


Now, we continue with the S-polynomials of a second type.

S(qi+m,i+1, qi+m+s,i+s+1) = (xi+1 · · ·xi+s) · qi+m,i+1 − (xi+m+1 · · ·xi+m+s) · qi+m+s,i+s+1

= lt(pi+1,s+1) · qi+m,i+1 − lt(pi+m+1,s+1) · qi+m+s,i+s+1

for all 0 ≤ i ≤ m− 3 and 1 ≤ s ≤ m− i− 2. By Proposition 2.16 (i) with i = i+ s+ 1, s = m− s− 1,

and t, j = m− 1 we get that

pi+1,s+1 · qi+m,i+1 − pi+m+1,s+1 · qi+m+s,i+s+1 = pi+2,s · qi+m,i − pi+m+1,s · qi+m+s+1,i+s+1

Thus

S(qi+m,i+1, qi+m+s,i+s+1) = −red(pi+1,s+1) · qi+m,i+1 + red(pi+m+1,s+1) · qi+m+s,i+s+1

+pi+2,s · qi+m,i − pi+m+1,s · qi+m+s+1,i+s+1

for all 0 ≤ i ≤ m − 3 and 1 ≤ s ≤ m − i − 2. Therefore, S(qi+m,i+1, qi+m+s,i+s+1) →B0 0. We finish

with the S-polynomials of a third type.

S(qi,i+m, qi′+m,i′+1) =

(xi′+m+1 · · ·xi) · qi,i+m − (xi′+1 · · ·xi+m−1) · qi′+m,i′+1 if i ≤ i′ − 1,

(xi′+m+1 · · ·xi′) · qi,i+m − (xi+1 · · ·xi+m−1) · qi′+m,i′+1 if i = i′,

(xi′+m+1 · · ·xi′) · qi,i+m − (xi+1 · · ·xi+m−1) · qi′+m,i′+1 if i = i′ + 1,

(xi+m · · ·xi′) · qi,i+m − (xi+1 · · ·xi′+m) · qi′+m,i′+1 if i ≥ i′ + 2,

=

lt(pi′+m+1,m+i−i′+1) · qi,i+m − lt(pi′+1,m+i−i′) · qi′+m,i′+1 if i ≤ i′,

lt(pi+m,m+i′−i+2) · qi,i+m − lt(pi+1,m+i′−i+1) · qi′+m,i′+1 if i ≥ i′ + 1,

for all 0 ≤ i ≤ m− 1 and 0 ≤ i′ ≤ m− 2. As in the previous cases, by Proposition 2.16 we get that

pi′+m+1,m+i−i′+1 · qi,i+m − pi′+1,m+i−i′ · qi′+m,i′+1 = pi′+m+1,m+i−i′ · qi+1,i+m − pi′+2,m+i−i′−1 · qi′+m,i′ ,

pi+m,m+i′−i+2 · qi,i+m − pi+1,m+i′−i+1 · qi′+m,i′+1 = pi+m+1,m+i′−i+1 · qi,i+m−1 − pi+1,m+i′−i · qi′+m+1,i′+1.

Finally, in a similar way as in the previous types of S-polynomials, it is not difficult to see that using

these identities that S(qi,i+m, qi′+m,i′+1) can be reduced to zero by B0 for all 0 ≤ i ≤ m − 1 and

0 ≤ i′ ≤ m− 2. For instance, if i = i′, i′ + 1, then

S(qi,i+m, qi′+m,i′+1) = red(pi,i+m+1) · qi′+m,i′+1 − red(pi′+m,i′+2) · qi,i+m.

Remark. Note that Theorems 2.17 and 2.19 are independent of the base ring.

We present the case of the cycle with four vertices.

Example 2.20. Let n = 4 = 2m and B0 as in Theorem 2.19. It is not difficult to compute that

q1,3 = x2 + x4, q2,4 = x1 + x3, q1,2 = x3x4, and

S(q1,3, q2,4) = x4 · q2,4 − x3 · q1,3 with lt(x4 · q2,4) = x1x4 6= x2x3 = lt(x3 · q1,3),

S(q1,3, q1,2) = x4 · q1,2, and S(q2,4, q1,2) = x3 · q1,2.

Therefore B0 is a Grobner basis for I3(C4).


Now, we present the cycle with six vertices.

Example 2.21. Let n = 2m = 2(3) = 6, then the polynomials q6,3 = x1x2 + x4x5 − 2, q1,4 =

x2x3 + x5x6 − 2, q2,5 = x3x4 + x1x6 − 2, q3,1 = x4x5x6 − x4 − x6 + x2, q4,2 = x1x5x6 − x1 − x5 + x3

(the leading terms are underlined) form a Grobner basis for I5(C6, X) with respect to the graded

lexicographic order with x2 > x3 > x4 > x5 > x6 > x1. For instance, taking i = 0 and s = m− 1 = 2

it is not difficult to see that

S(q6,3, q2,5) = x3x4 · q6,3 − x1x2 · q2,5 = (x4x5 − 2) · q2,5 − (x5x6 − 2) · q3,6.

Also, taking i = 1 and s = 1 we get that

S(q1,4, q2,5) = x4 · q1,4 − x2 · q2,5 = x4x5x6 − 2x4 − x2x1x6 + 2x2 = q3,1 − q3,5 = 2q3,1 − x6 · q6,3.

In order to finish this chapter we present a simple application of Theorem 2.11, see Remark 6.8 in [32]

for a similar result for graphs.

Corollary 2.22. Let D be a digraph and v ∈ V (D) such that D \ v ' Cn, then K(D) has at most two

invariant factors different to one.

It is not easy to determine when the critical group of G is cyclic because we need to evaluate a set of

polynomials and after to compute its greatest common divisor.

Chapter III : Critical ideals of trees

This chapter focuses mainly to present an explicit description of a set of generators for the critical

ideals when G is a tree. Let T be a tree, that is, a simple graph without cycles, and put T ` as the

(non-simple) graph with V (T `) = V (T ) and E(T `) = E(T )∪{vv|v ∈ V (T `)}. Our description is based

on the set of 2-matchings of T `. More precisely: To each 2-matching M on T ` , we associate a minor

of L(T,X), denoted d(M, X), in such a way that if |M| = j then d(M) is a j-minor. This leads to

our main result.

Theorem III.3.6. If T is a tree with n vertices, then

Ij(T,X) =⟨d(M, X)

∣∣M∈ V∗2 (T `, j)⟩

for all 1 ≤ j ≤ n.

Here V∗2 (T `, j) denote a special set of 2-matchings whose definition is given in section 2.

When one is intending to describe the critical group of a particular graph (as in any finite group)

getting information about his rank is important. If we set γ(G) = max{j|Ij(G,X) = Z[XG]}, then the

first γ(G) invariant factors of K(G) are equal to one and therefore rank(K(G)) ≤ n− 1− γ(G).

The invariant γ have some nice features. For example, Lemma II.1.10 asserts that γ(G) ≤ min(2n −α(G), 2n−ω(G)−1), where α(G) and ω(G) are respectively the stability and the clique numbers of G.

A remarkable consequence of Theorem 2.6 is the characterization of γ(T ). Lett ν2(G) be the maximum

size of a 2-matching on G.

Theorem III.3.8. If T is a tree, then γ(T ) = ν2(T ),

This led us to prove Conjecture II.2.10,

Corollary III.3.9. If G is a simple graph in n vertices, then γ(G) = n− 1 if and only if G = Pn.

In order to stablished this results we present first the concept of 2-matching and develop their prop-

erties that will be used after. This is the content of Section 2. Section 3 begins by establishing the

correspondence between 2-matchings on T ` and minors of L(T,X). For the importance that this rela-

tion has in our results we illustrate it with several examples. After doing this we focus in the algebraic

relations between the minors associated to 2-matchings. On section 4 we put special attention to the

case j = n− 1. It turns that,

50 Chapter III : Critical ideals of trees

Theorem III.4.9. B = {d(M, X)|M ∈ V∗2 (T, n− 1)} is a reduced Groebner basis for In−1(T,X).

Section 5 is devoted to present 3 examples about how our results can be used to obtain concrete

information about critical groups. First we present arithmetical trees associated to the reduction of

elliptic curves of Kodaira type I∗n. Second we study the critical ideals of the graph obtained from a

regular tree by collapsing the leaves to a single vertex and reply some results obtained by Levine [27]

and Toumpakary [36]. Finally we describe critical ideals of trees with depth two.

1. 2-matchings of trees

In this section we introduce the concept of 2-matching of a graph. After that, we present some of the

properties of a 2-matching of a tree, which will be very useful to give a description of the critical ideals

of a tree.

We begin by defining what means a 2-matching of a graph.

Definition 1.1. Let G be a graph (possibly with loops and multiple edges) and M a set of the edges of

G. We say that M is a 2-matching if every vertex of G has at most two incident edges in M.

The set of all 2-matchings of a graph G, will be denoted by V2(G). Moreover, let V2(G, j) be the set

of 2-matchings of G of size j, that is, with j edges. Also, the 2-matching number of G, denoted by

ν2(G), is the maximum number of edges of a 2-matching of G. A 2-matching of G of size ν2(G) is

called maximum.

The concept of 2-matching apply for any class of graphs, however in this article we are primarily

interested when G is a tree. Let T be a tree, that is, a graph without cycles. Is not difficult to see,

that a 2-matching of a tree consist of a union of paths, we recall that a vertex is a path of length zero.

As a example, let C be the tree given in Figure III.1.a. If we take (see Figure III.1.b and III.1.c)

M1 = {v1v2, v2v5, v6v7, v6v8} and M2 = {v1v2, v2v3, v2v4, v6v8}.

Then M1 ∈ V2(C, 4) and M2 6∈ V2(C) becuase M2 has 3 incident edges on v2.

v1 v2

v3

v4

v5 v6

v7

v8

v9 v1 v2

v3

v4

v5 v6

v7

v8

v9 v1 v2

v3

v4

v5 v6

v7

v8

v9

(a) A caterpillar tree C (b) A 2-matching of C (c) A no 2-matching of C

Figure III.1: A caterpillar tree without two pairs of legs.

Is important to note that a loop vv is counted twice as an incident edge of v, however it is only one

edge in the 2-matching. Now, we focus our attention in an special set of the 2-matchings, the maximal

ones.

Section 1 : 2-matchings of trees 51

Definition 1.2. A 2-matching M of a graph G is called maximal if not exists a 2-matching N such

that M ( N .

For instance, the continuos edges in the tree given in the center of Figure III.1 is a maximal 2-matching.

This maximal 2-matching will play an important role in the description of the critical ideals of a tree.

In the follow we present the first property of the maximal 2-matching of a tree.

Given two vertices u, v of a tree T , let Pu,v be the unique path in T that join u and v.

Proposition 1.3. If M is a maximal 2-matching of a tree T , then exists u 6= v leaves of T such that

E(Pu,v) ⊆M.

Proof. The proof follows by induction on the number of vertices of T . Is clear that the result is true

for the trees with less or equal to three vertices.

Now, assume that the result is true for all the trees with k or less vertices. Let T be a tree with k + 1

vertices,M be a maximal 2-matching of T , a a leaf of T and e = ab be the edge of T incident with a. If

e /∈M, thenM is a maximal 2-matching of T \a and the result it follows by the induction hypothesis.

On the other hand, if e ∈ M, then M\ e is a maximal 2-matching of T \ a. By induction hypothesis,

let u 6= v leaves of T \ a such that E(Pu,v) ⊆M \ e. If b 6= u, v, then the result it follows. Otherwise,

if b = u, then a and v are leaves of T such that E(Pu,v) ⊆M.

In the following we explore the 2-matching number of a tree when we delete one of its edges. Before

to present the result, we introduce some concepts. Given a tree T and a vertex v, we say that v is

saturated if any maximum 2-matching of T has two incident edges to it. In a similar way, we say that

an edge e of T is saturated when it appears in all the maximum 2-matchings of T .

Lemma 1.4. If T is tree without loops and e = uv is an edge of T , then

ν2(T )− ν2(T \ e) =

0 if and only if u or v are saturated in T \ e,

1 if and only if e are saturated in T.

Proof. Let Tu and Tv be the connected components of T \e that contain the vertex u and v respectively.

Let M be a maximum 2-matching of T , Mu = M ∩ E(Tu) and Mv = M ∩ E(Tv). Note that

Mu and Mv are not necessarily maximum 2-matchings of Tu and Tv respectively. However, we can

assure that at least one of them if it is and the other is almost maximum (size ν2(Tv) − 1). Since

|M| = |Mu|+ |Mu|+ |M∩ {e}| and 0 ≤ |M∩ {e}| ≤ 1, then ν2(T ) ≤ ν2(T \ e) + 1. In a similar way,

taking maximum 2-matchings of Tu and Tv we get that ν2(T \ e) ≤ ν2(T ).

Now, e is not saturated in T if and only if there exist Nu and Nv maximum 2-matchings of Tu and

Tv respectively such that N = Nu ∪ Nv is maximum of T if and only if ν2(T ) = ν2(T \ e). That is,

ν2(T ) = ν2(T \ e) + 1 if and only if e is saturated in T .

On the other hand, e is saturated in T if and only if each maximum 2-matching M of T satisfies that

e ∈ M and M\ e is a maximum 2-matching of T \ e if and only if degT [M\e](u), degT [M\e](v) ≤ 1 if

and only if u and v are not saturated in T \ e.

Now, we see how it changes the 2-matching number of a tree when we delete one if its vertices.


Lemma 1.5. Let T be a tree without loops, v be a vertex of T , NT (v) = {w1, . . . , ws}, and Ti the

connected component of T \ v that contains wi. Then

ν2(T )− ν2(T \ v) =

2 if and only if v is saturated in T,

1 if and only if there exist 1 ≤ j ≤ s such that vwj is saturated and

wi is saturated in Ti for all i 6= j,

0 if and only if wi is saturated in Ti for all wi ∈ NT (v).

Proof. Given a maximum 2-matching M of T , let Mi =M∩ E(Ti). Note that Mi is not necessarily

a maximum 2-matchings of Ti. However, we can assure that in the following cases: when vwi /∈ Mor vwi ∈ M but v is saturated. Assume that Mi is not a maximum 2-matching of Ti, that is, there

exists a maximum 2-matchings M′i of Ti with |M′i| > |Mi|. If vwi /∈ M, then M′ = (M\Mi) ∪M′iis a 2-matching of T with |M′| > |M|; a contradiction. Also, if v is saturated and vwi ∈ M, then

M′ = [M\(Mi∪{vwi})]∪M′i is a 2-matching of T with |M′| ≥ |M| and degM′(v) = 1; a contradiction.

Since T \ v = T1 t · · · t Ts, then ν2(T \ v) = ν2(T1) + · · ·+ ν2(Ts) and

2 ≥ |M∩ δT (v)| = |M \( s⋃i=1

Mi

)| ≥ ν2(T )−

s∑i=1

ν2(Ti) = ν2(T )− ν2(T \ v) = ν2(T )− ν2(T \ v),

where δT (v) = {vw | vw ∈ E(T )}. That is ν2(T )−ν2(T \v) ≤ 2. On the other hand, clearly ν2(T \v) ≤ν2(T ) and therefore ν2(T \ v) ≤ ν2(T ) ≤ ν2(T \ v) + 2.

Now, if ν2(T )− ν2(T \ v) = 2, then |M ∩ δT (v)| = 2 and v is saturated in T . Also, if v is saturated in

T , then the Mi are maximum 2-matchings in Ti and

ν2(T ) = |M| =s∑i=1

|Mi|+ 2 =s∑i=1

ν2(Ti) + 2 = ν2(T \ v) + 2.

Also, is not difficult to check that: ν2(T ) = ν2(T \ v) if and only if wi is saturated in Twi for all

wi ∈ NT (v). Finally, if ν2(T ) = ν2(T \ v) + 1, then there exist 1 ≤ j ≤ s such that vwj is saturated

and wi is saturated in Twi for all wi ∈ NT (v) \ wj . For the converse, we have the following

ν2(T \ v) =

s∑i=1

ν2(Ti)wi is saturated ∀ i 6=j

= ν2(Tv) + ν2(Tj) = ν2(T \ vwj)1.4= ν2(T )− 1,

where Tv is the connected component of T \ vwj that contains v.

Next result prove that if wi is saturated in Ti, then wi is saturated in T . Is not difficult to check that

the converse is not true.

Proposition 1.6. Let T be a tree without loops, uv ∈ E(T ), and Tu be the connected component of

T \ uv that contains u. If u is saturated in Tu, then u is saturated in T .

Proof. Let Tv be the connected component of T \ uv that contains v. Since u is saturated in Tu and

T \ e = Tu t Tv, then by Lemma 1.4, ν2(T ) = ν2(Tu) + ν2(Tv). Thus

ν2(T ) + ν2(T \ u) = ν2(Tu) + ν2(Tv)− [ν2(Tu \ u) + ν2(Tv)] = ν2(Tu)− ν2(Tu \ u) = 2.

Finally, by Lemma 1.5, u is saturated in T .

Section 1 : 2-matchings of trees 53

Corollary 1.7. If T is a tree with at least three vertices, then there exist at least one vertex v of T

such that ν2(T ) = ν2(T \ v) + 2.

Proof. It follows by Lemma 1.5 and Proposition 1.6. The only tree that not satisfies this theorem is

the tree with only one edge.

Now, we will study the 2-matchings of graphs obtained by adding a loop in each of its vertices.

Two matchings of G`

Given a simple graph G, let G` be the graph obtained from G by adding a loop on each vertex of G.

That is, E(G`) = E(G) ∪ {uu |u ∈ V (G)}. Given M ∈ V2(G`), let `(M) = M∩ {uu |u ∈ V (G)}.Now, we introduce the concept of minimal 2-matching, which is central in the description of the critical

ideals of a tree.

Definition 1.8. A 2-matching M of G` is called minimal if not exists a 2-matching M′ such that

`(M′) ( `(M) and |M| = |M′|.

Directly from the definition we get that: if M ∈ V2(G`) is minimal, then |M| > ν2(G). Also, some

maximal 2-matching of T is part of a minimal 2-matching of T ` as next result show.

Proposition 1.9. If M is a maximal 2-matching of T [NT (M)], then

N =M∪ {uu |u /∈ V (M)}

is a minimal 2-matching of T `.

Proof. Assume that N is a not minimal 2-matching of T `. Thus, there exist a 2-matching N ′ of T `

such that |N | = |N ′| and `(N ′) ( `(N ). That is, N ′ has at least one more edge as N . Since M is

maximal on NT (M), N ′ must have an edge with at least one end in U ; a contradiction to the fact that

N has a loop in all the vertices of U .

Example 1.10. Let C be the caterpillar tree consider in Figure III.1.a.

v1 v2

v3

v4

v5 v6

v7

v8

v9 v1 v2

v3

v4

v5 v6

v7

v8

v9 v1 v2

v3

v4

v5 v6

v7

v8

v9 v1 v2

v3

v4

v5 v6

v7

v8

v9

(a) (b) (c) (d)

Figure III.2: A caterpillar tree C and some of its minimal 2-matchings.

Is not difficult to check that ν2(C) = 4. Thus, any minimal 2-matching of C` has at least size 5. The

2-matchingM1 = {v1v2, v2v5, v5v6, v6v9, v3v3}, see Figure III.2.a, is a minimal 2-matching of C` of size


5 with only one loop. Also, the 2-matching given in Figure III.2.b is a minimal 2-matching of C` of size

6.

LetM3 = {v1v1, v3v2, v2v4, v5v5, v6v6, v7v7, v8v8, v9v9} be the 2-matching given in Figure III.2.c. Using

Proposition 1.9 is not difficult to check that M3 is a minimal 2-matching of size 8. Moreover, M4 =

M3 \ {v1v1} (see Figure III.2.d) is also a minimal 2-matching of size 7.

Now, we give a description of all the minimal 2-matchings on T `.

Proposition 1.11. Let V∗2 (G) be the set of minimal 2-matchings of G. If T is a tree and e = uv ∈E(T ), then

V∗2 (T `) ⊆ {V∗2 (T ù + e)e ∪ V∗2 (T `v + e)e} ∪ {V∗2 (T ù) ∪ V∗2 (T `v )},

where Tx is the subtree of T \ e that contains the vertex x and V∗2 (G)e is the set of 2-matchings of G

that contain the edge e.

Proof. Let M be a minimal 2-matching on T `. Suppose that e ∈ M and let Mu =M∩ (Tu + e) and

Mv =M∩ (Tv + e). As M =Mu ∪Mv is enough to prove that Mu is minimal on T ù + e. Suppose

the opposite, then there exist M′u ∈ V2(T ù + e) such that `(M′u) ( `(Mu) and |M′u| = |Mu|. Note

that `(M′u ∪Mv) ( `(M) and

|M′u ∪Mv| =

|M|+ 1 if e 6∈ M′u,

|M| if e ∈M′u.

Thus, given that M is minimal, e 6∈ M′x and |M′u ∪ Mv| = |M| + 1. If we remove one loop (or

an edge diferent from e) of M′u, then we get a 2-matchings M′′u such that |M′′u ∪ Mv| = |M| and

`(M′′u ∪Mv) ( `(M) which also contradict the minimality ofM. Thus,Mu is minimal on Tu + e. As

e ∈Mu,Mv, M∈ V∗2 (T ù + e)e ∪ V∗2 (T `v + e)e.

Now, suppose that e 6∈ M. Let Mu =M∩ Tu and Mv ∩ Tv. The minimality of Mu and Mv can be

deduced in a similar way.

2. Critical Ideals of Trees

This section is devoted to establish a relationship between the generators of the critical ideals and the

2-matchings of a given tree. This relationship allows to give a complete combinatorial description of

the critical ideals of a tree T . More precisely, each 2-matching has associated one generator of their

critical ideals. Moreover, we prove that the critical ideals are generated from an special set of the

2-matching of the tree.

Since the j critical ideal of a graph G is generated by the j-minors of their generalized Laplacian

matrix, then it only depends of the non-vanishing j-minors of L(G,XG). Therefore, we begin by giving

a description of the non-vanishing j-minors of the generalized Laplacian matrix of a tree.

Section 2 : Critical Ideals of Trees 55

The non-vanishing minors of L(T,XT ).

In this section we prove that the non-vanishing j-minors of L(T,XT ) correspond to the 2-matchings of

T . Assume that T has n vertices and we impose an order on their vertices. We begin by introducing

some notation.

Given a 2-matching M of T `, we associate the sets t(M), h(M) ⊂ V (T ) as follows: First, if M =

{vj1vj2 , vj2vj3 , . . . , vjmvjm+1} is connected, then

h(M) = {vj2 , . . . , vjm+1} and t(M) = {vj1 , . . . , vjm}.

That is, if−→M is the oriented path obtained from M and the order on the vertices of T , then h(M)

and t(M) are the heads and tails of their arcs, respectively. Moreover, if M is non-connected and

{M1, . . . ,Mk} are their connected components, then h(M) = ∪ms=1h(Mi) and t(M) = ∪ms=1t(Mi).

On the other hand, let L(T,X)[t(M), h(M)] be the square submatrix of L(T,X) of size |M| =∣∣t(M)

∣∣ =∣∣h(M)∣∣, aM the leading coefficient of det (L(T,X)[t(M), h(M)]), and

d(M, X) =

{det (L(T,X)[h(M), t(M)]) if aM > 0,

−det (L(T,X)[h(M), t(M)]) if aM < 0.

Thus d(M, X) is a generator of the |M|-critical ideal of T . As we will see in Lemma 2.2, d(M, X) does

not depend on the order of the vertices of T . That is, the correspondence M 7−→ d(M, X) between

V2(T `) and Z[XG] is well defined. Next example illustrate this correspondence between 2-matchings

and generators of the critical ideals of T .

Example 2.1. Let T be the tree given in Figure III.1. Is not difficult to see that

M = {v1v1, v2v3, v2v5, v6v7, v6v8, v9v9}

is a 2-matching of T . Moreover, the paths P1 = v3v2v5 and P2 = v7v6v8 and the loops L1 = v1v1

and L2 = v9v9 are the connected components of M. Since h(P1) = {v2, v5}, t(P1) = {v3, v2},h(P2) = {v6, v8}, t(P2) = {v7, v6}, h(L1) = {v1} = t(L1), and h(L2) = {v1} = t(L2), then h(M) =

{v1, v2, v5, v6, v8, v9} and t(M) = {v1, v2, v3, v6, v7, v9}. Thus

L(T,X)[h(M), t(M)] =

x1 −1 0 0 0 0

−1 x2 −1 0 0 0

0 −1 0 0 0 0

0 0 −1 x6 −1 −1

0 0 0 −1 0 0

0 0 0 −1 0 x9

and d(M, X) = x1x9.

Lemma 2.2. If T is a tree and M a 2-matching of T `, then

d(M, X) = x`(M) + ”terms of lower degree”.


Proof. First, is not difficult to prove that if P is a disjoint union of paths in a tree, then

L(T,X)[h(P), t(P)] ∼

1 ∗ ∗

0. . . ∗

0 0 1

.

Thus

L(T,X)[h(M), t(M)] ∼

L(T,X)[`(M), `(M)] ∗ ∗ ∗

0 1 ∗ ∗

0 0. . . ∗

0 0 0 1

,

where `(M) = {u1, . . . , ur} is the set of loops of M . Since

L(T,X)[`(M), `(M)] =

xu1 ∗ ∗

∗ . . . ∗∗ ∗ xur

and det (L(T,X)[h(M), t(M)]) = det (L(T,X)[l(M), l(M)]), then the result is clear.

The next lemma is a partial converse of the previous result.

Lemma 2.3. If f(X) is a non-vanishing minor of L(T,X) with positive leader coefficient, then there

exist M∈ V2(T `) such that f(X) = d(M, X).

Proof. Let I, J ⊆ V (T ) such that |I| = |J | 6= 0 and f(X) = det(L(T,X)[I, J ]

). Since f(X) is non-

zero, we can assume that all the entries in the main diagonal of L(T,X)[I, J ] are different from zero.

Now, let

N = {vi1vj1 , . . . , vitvjt},

where I = {i1, . . . , it} and J = {j1, . . . , jt}. Since ir 6= is and jr 6= js for all r 6= s, then N is a

2-matching of T ` with |N | ≤ t = |I|. If |N | < t, then exists 1 ≤ r < s ≤ t such that virvjr = visvjs .

Since ir 6= is, ir = js, jr = is and N ′ = N ∪ {virvjs , visvjr} \ {virvjr , visvjs}, is a 2-matching N ′ of

T ` with |N ′| = |N | + 1. We can repeat this process until to get a 2-matching M of size t such that

I = t(M) and J = h(M). That is, d(M, X) = f(X).

By Lemma 2.3, Ij(T,X) = 〈d(M, X) |M ∈ V2(T `) with |M| = j〉 for any tree T with n vertices and

1 ≤ j ≤ n. However, this description is not minimal. For instance, is not difficult to find a tree T

and 2-matchings M 6= N of T ` with `(M) = `(N ). That is, d(M, X) = d(N , X) (Lemma 2.2) and

therefore the previous description of Ij(T,X) contains repeated elements. Moreover, the minors of

L(T,X) are related by more complex algebraic identities.

In the next we exploit the combinatorial structure of T in order to develop some identities between the

minors of L(T,X) which allows to find a better description of the critical ideals of a tree. Before to

present the first of this identities, we fix some notation. For any graph G, let d(G,X) = det(L(G,X)).

Section 2 : Critical Ideals of Trees 57

Lemma 2.4. If T is a tree and S ⊆ E(T ), then

d(T \ S,X) =∑

µ∈V1(S)

d(T \V (µ), X),

where V1(S) is the set of matchings of the subgraph of T induced by S.

Proof. We use induction on |S|. First, let S = {uv}. Since

V1(T ) = V1(T \ uv)⋃{{uv} ∪ µ

∣∣µ ∈ V1(T \ {u, v})}

and d(T,X) =∑

µ∈V1(T )(−1)|µ|∏v 6∈V (µ) xv, ([20, Lemma 4.4]) we get the result, that is, d(T \uv,X) =

d(T,X) + d(T \ {u, v}, X).

Now, let S = {uv} ∪ S′ with |S′| > 0. If T ′ = T \ S′, then by induction hypothesis

d(T \ S,X) = d(T ′ \ uv,X) = d(T ′, X) + d(T ′ \ {u, v}, X)

and d(T ′, X) = d(T \S′, X) =∑

µ∈V1(S′) d(T \V (µ), X). On the other hand, since T ′\{u, v} = Tu,v\S′′,where Tu,v = T \ {u, v} and S′′ = {e ∈ S′ |u, v 6∈ V (e)},

d(T ′ \ {u, v}, X) = d(Tu,v \ S′′, X) =∑

µ∈V1(S′′)

d(Tu,v \ V (µ), X).

Moreover, since V1(S) = V1(S′)⋃{{uv} ∪ µ

∣∣µ ∈ V(S′′)}

,

d(T \ S,X) =∑

µ∈V1(S′)

d(T \ V (µ), X) +∑

µ∈V1(S′′)

d(Tu,v \ V (µ), X) =∑

µ∈V1(S)

d(T \V (µ), X).

This lemma is a fundamental result on this paper, in the sense that almost all the identities between the

generators of the critical ideals of a tree are derived from it. For instance we have the next corollary:

Corollary 2.5. If M is a 2-matching of T and w a vertex such that ww /∈ `(M), then

xw d(M, X) = d(N , X) +∑v∈U

d(M\ {vv}, X),

where N = {uv |uv ∈M and w 6= u, v} ∪ {ww} and U = {v ∈ V (T ) | vv ∈ `(M), vw ∈ E(T )}.

Proof. Let T ′ = T [`(N )] and S be the set of edges in T ′ that contains w. Since V1(S) = ∅ ∪ {vw | vv ∈`(M), vw ∈ E(T )}, then applying Lemma 2.4 to T ′ and S we get that

d(T ′ \ S,X) =∑

µ∈V1(S)

d(T ′ \V (µ), X) = d(T ′, X) +∑v∈U

d(T ′ \{w, v}, X).

Since w and T ′ \ S are not connected, then d(T ′ \ S,X) = xwd(T ′ \ w,X) = xwd(M, X). On the

other hand, by Lemma 2.2, d(T ′, X) = d(N , X) and d(T ′ \{w, v}, X) = d(M\ {vv}, X) for all v ∈ U .

Combining this identities we get the result.


This corollary allows us to prove one of the most important results in this article.

Theorem 2.6. If T is a tree with n vertices, then

Ij(T,X) =⟨{

d(M, X) |M ∈ V∗2 (T `, j)}⟩

for all 1 ≤ j ≤ n.

Proof. By Lemma 2.3, Ij(T,X) ⊆ {d(M, X)∣∣M ∈ V2(T `, j)}. Thus, we only need to prove that

the minor of a not minimal 2-matching can be expressed in term of minors associated to minimals

2-matchings of the same size.

LetM be a not minimal 2-matching of size j. Then, there exist N ∈ V2(T, j) and w ∈ V (T ) such that

`(M) = `(N ) ∪ {ww}. Applying Proposition 2.5 to N we get that

d(M, X) = xw d(N , X) −∑v∈U

d(N \ {vv}, X),

where U = {v ∈ V (T ) | vv ∈ `(N ), vw ∈ E(T )}.For all v ∈ U , let Nvw = (N \ {vv}) ∪ {vw}. Since vw 6∈ N \ {vv}, clearly |Nvw| = |N \ {vv}| + 1

and therefore Nvw is 2-matching of T ` of size j. On the other hand, since `(Nvw) = `(N ) \ {vv}, by

Lemma 2.2 we get that d(N \ {vv}, X) = d(Nuv, X). Therefore

d(M, X) = xwd(N , X) −∑v∈U

d(Nvw, X).

Since `(N ) ( `(M) and `(N )vw ( `(M) for all v ∈ U , we can repeat this process until we get

an expression of d(M, X) as a algebraic combination on the minors associated to some minimal 2-

matchings of T ` of size j.

Next example illustrate how it works the previous theorem.

Example 2.7. Let C be the tree given in Figure III.1 and

M = {v1v1, v2v2, v3v3, v4v4, v5v5, v6v6}

be a 2-matching of C` of size 6. Since M1 = {v1v1, v2v2, v3v3, v4v4, v5v5, v6v9} is a 2-matching of size

6 with `(M) = `(M1) ∪ {v6v6}, then M is non-minimal. Thus d(M, X) = x6d(M1, X) − d(M2, X),

where M2 = {v1v1, v2v2, v3v3, v4v4, v5v6, v6v9}.In a similar way, since M1 and M2 are not minimal, then d(M1, X) = x5d(M2, X) − d(M3, X)

whereM3 = {v1v1, v2v5, v3v3, v4v4, v5v6, v6v9} and d(M2, X) = x2d(M3, X)−d(M4, X)−d(M5, X)−d(M6, X) withM4 = {v1v2, v3v3, v4v4}∪P,M5 = {v1v1, v2v3, v4v4}∪P,M6 = {v1v1, v2v4, v3v3}∪P,

and P = {v2v5, v5v6, v6v9}.Finally, since M4,M5 and M6 are minimal 2-matchings and d(M3, X) = x1d(M4, X), then

d(M, X) = (x1 · p2,5,6−p5,6) · d(M4, X)−p5,6 · d(M5, X)−p5,6 · d(M6, X),

where p2,5,6 = x2x5x6−x2−x6 and p5,6 = x5x6−1. In a similar way, we can get that

d(M, X) = (x4 · p2,5,6−p5,6) · d(M6, X)−p5,6 · d(M4, X)−p5,6 · d(M5, X),

= (x3 · p2,5,6−p5,6) · d(M5, X)−p5,6 · d(M4, X)−p5,6 · d(M6, X),

Section 3 : Groebner basis of critical ideals 59

which gives us an expressing of d(M, X) in terms of minors associated to some minimal 2-matchings

of C` of size 6.

Next result is fundamental identity in the study of the critical ideals of trees, which prove that the first

non trivial critical ideal of a tree is the ν2(T )-critical ideal.

Theorem 2.8. If T is a tree, then γ(T ) = ν2(T ).

Proof. Since Iν2(T )(T,X) is trivial we only need to prove that Iν2(T )+1(T ) is non-trivial. We will use

induction in the number of vertices of the tree. It is not difficult to check that the result is true for all

the trees with less or equal to four vertices, therefore we can assume that |V (T )| ≥ 5. Let k = ν2(T )+1

and v ∈ V (T ). By [20, Claim 3.12]

Ik(T,X) ⊆ 〈xvIk−1(T \ v,X), Ik−2(T \ v,X), Ik(T \ v,X)〉 .

Moreover, since Ik(T \ v,X) ⊆ Ik−1(T \ v,X) ⊆ Ik−2(T \ v,X), then Ik(T,X) ⊆ 〈xv, Ik−2(T \ v,X)〉.By induction hypothesis γ(T \ v) = ν2(T \ v) for all v ∈ V (T ). If we assume that Ik(T,X) is trivial,

then Ik−2(T \ v,X) is trivial and therefore

ν2(T )− 1 = k − 2 ≤ γ(T \ v) = ν2(T \ v) for all v ∈ V (T );

a contradiction to Lemma 1.4.

As a consequence we get that Pn is the only one simple graph with n vertices and γ(G) = n− 1.

Corollary 2.9. If G is a simple graph in n vertices, then γ(G) = n− 1 if and only if G = Pn.

Proof. (⇒) It is clear from Theorem 2.6. (⇐) Let G be a graph with n vertices and γ(G) = n − 1.

Since In−1(G,X) = 〈1〉, by [20, Proposition 3.7] the critical group of G must be trivial. Then by the

Kirchhoff’s Matrix Tree Theorem [11, Theorem 6.2], G is a tree. Finally, by Theorem 2.8 we get that

ν2(G) = n− 1 and therefore G is the path with n vertices.

3. Groebner basis of critical ideals

In this section we prove that if T is a tree with n vertices, then {d(M, X)|M ∈ V∗2 (T `,m− 1)} is not

only a generating set this is also a Groebner basis for In−1(T ). First we prove that for any 1 ≤ j ≤ n

strong reduction by V2(T `, j) is equivalent with strong reduction by V∗2 (T `, j).

Proposition 3.1. Let 1 ≤ j ≤ n and XT = {x1, . . . , xs}. If f(x), g(x) ∈ Z[XT ] are such that

f(x)→V2(T `,j) g(x), then f(x)→V∗2 (T `,j) g(x).

Proof. Suppose that d(M, X) ∈ V2(T `, j) and h(x) ∈ Z[XT ] are such that g(x) = f(x)− h(x)d(M, X)

and xg ≺ xf . If M is minimal, then there is nothing left to prove. On the other hand, if M is not

minimal, then according to Theorem 2.6 there exist N1, . . . ,Ns ∈ V∗2 (T `, j) and t1(x), . . . , ts(x) ∈ Z[XT ]

such that d(M, X) = t1(x)d(N1, X) + · · ·+ ts(x)d(Ns, X). Thus

g(x) = f(x)−s∑i=1

ti(x)p(Ni, X)h(x).


Following the proof of Theorem 2.6 we can ensure that for each i = 1, . . . , s− 1

lt(ti(x)p(Ni, X)) ≺ lt(ti+1(x)p(Ni+1, X)).

Thus, if

f1(x) = f(x)− t1(x)p(N1, X)h(x),

f2(x) = f1(x)− t2(x)p(N2, X)h(x),

...

fs(x) = fs−1(x)− ts(x)p(Ns, X)h(x),

then xfs ≺ · · · ≺ xf1 ≺ xf . Therefore f(x)→V∗2 (T `) f1(x)→V∗2 (T `) · · · →V∗2 (T `) fs(x) = g(x).

Now, before to proceed to deal with the reduction of S-polynomials, we begin with the reduction of a

monomial and a minor of size n − 1. In the next, if e1, e2 are two different edges in T , then P (e1, e2)

is the unique path in T that join e1 and e2.

Lemma 3.2. If T is a tree and P is a not empty path of T , then

xP d(T \ P,X) = d(T,X) +∑

e∈E(NT (P ))

d(T \ V (e), X) +∑

(e1,e2)∈Λ

xP (e1,e2)

xe1xe2d(T \ P (e1, e2), X),

where Λ = {(e1, e2) ∈ V1(NT (P ))|e1, e2 ∈ E(NT (P ))}.

Proof. Let S = E(NT (P )). Clearly V (P ) is a free set of T \ S. Thus, by Lemma 2.4

xP d(T \ P,X) =∑

µ∈V1(S),|µ|≤1

d(T \ V (µ), X) +∑

µ∈V1(S),|µ|≥2

d(T \ V (µ), X).

Each µ ∈ V1(S) with |µ| = 2 is a member in Λ. If Eµ is the neighborhood of V (P (µ))/V (µ) in

T [V (P (µ))/V (µ)], then{µ ∈ V1(NT (P ))

∣∣|µ| ≥ 2}

=⋃µ∈Λ

{µ ∪ ρ

∣∣ρ ∈ V1(Eµ)}

. This relation allow us

to write ∑µ∈V1(S),|µ|≥2

d(T \ V (µ), X) =∑µ∈Λ

∑ρ∈V1(Eµ)

d(T \ V (µ ∪ ρ), X).

For each µ ∈ Λ we apply Lemma 2.4 in T \ V (µ) and Eµ to get that∑ρ∈V1(Eµ)

d(T \ V (µ ∪ ρ), X) =xP (µ)

xµd(T \ P (µ), X).

Remark. Note that a 2-matching M has size n − 1 if and only if T \ `(M) is a path (possibly with

size 0). Thus, T [M] = T \ P for some path P and d(M, X) = d(T \ P,X). Conversely, for each path

P , T \ P = T [M] for some M∈ V2(T `, n− 1).

Section 3 : Groebner basis of critical ideals 61

Now we deal with other case of the product of a monomial and a minor of size n− 1. Suppose that P

and Q are non empty paths of T with Q ⊂ P . Then P \Q is composed of one or two paths, which we

call Pl and Pr (Pr could be empty). Let L = E(NT\Q(Pl)) and R = E(NT\Q(Pr)).

Proposition 3.3. Let P be a path in a tree T and Q a not empty subpath of P . If L and R are defined

as above, thenxPxQ

d(T \ P,X) = d(T \Q,X) +∑e∈L

x P (e,Q)

xexQd(T \ P (e,Q), X)

+∑e∈R

x P (Q,e)

xQxed(T \ P (Q, e), X) +

∑el∈L

∑er∈R

xP (el,er)

xelxQxerd(T \ P (er, Q, el), X).

Proof. Set T ′ = T \Q. As L ∪ R is the set of edges of NT ′(V (P ) \ V (Q)) and V (P ) \ V (Q) is free in

T ′ \ S = T \ P , by Lemma 2.4

xPxQ

d(T \ P,X) =∑

ν∈V1(L∪R)

d(T ′ \ V (ν), X).

For each e ∈ L let P (e,Q) be the path in T that join the vertices in e and Q and set Ve,Q = V (P (e,Q))\(V (e) ∪ V (Q)). If we set Se,Q = {uv ∈ E(T )|u, v ∈ Ve,Q}, then Se,Q is a set of edges on Te,Q =

T \ (V (e) ∪ V (Q)). Thus by Lemma 2.4,

d(Te,Q \ Se,Q, X) =∑

ν∈V1(Se,Q)

d(Te,Q \ V (ν), X).

Since, Te,Q \ Se,Q = T \ P (e,Q) ∪ Ve,Q and V1(L) = {∅} ∪e∈L {{e} ∪ V1(Se,Q)} thus∑ν∈V1(L)\{∅}

d(T ′ \ V (ν), X) =∑e∈L

∑ν∈V1(Se,Q)

d(T ′ \ V ({e} ∪ ν), X)

=∑e∈L

∑ν∈V1(Se)

d(Te,Q \ V (ν), X) =∑e∈L

xVe,Qd(T \ P (e,Q), X).

In the same way we get the expression that involve V1(R).

Set LR as the (non-empty) matchings on L∪R that intersect both L and R. For each el ∈ L and er ∈ Rlet P (el, er) be the only path that join el and er. If we set Vel,er = V (P (el, er))\ (V (el)∪V (Q)∪V (er))

and Sel,er = {uv ∈ E(T )|u, v ∈ Vel,er}, then Sel,er is a set of edges on Tel,er = T \ (V (el) ∪Q ∪ V (er)).

By Lemma 2.4

d(Tel,er \ Sel,er , X) =∑

ν∈V1(Sel,er )

d(Tel,er \ V (ν), X).

If we notice that Tel,er \Sel,er = T \P (el, er) +Ver,el , LR =⋃el∈L

⋃er∈R{{el, er}∪V1(Sel,er)} and that

for each ν ∈ V1(Sel,er), T′ \ V ({el, er} ∪ ν) = Tel,er \ V (ν), then we get∑

ν∈LRd(T ′ \ V (ν), X) =

∑el∈L

∑er∈R

∑ν∈V1(Sel,er )

d(Tel,er \ V (ν), X)

=∑e∈L

∑er∈R

d(Tel,er \ Sel,er , X) =∑e∈L

∑er∈R

xVel,erd(T \ P (e,Q), X).

This complete the proof of the theorem as V1(L ∪R) = V1(L) ∪ V1(R) ∪ LR.


The two last results are used to establish the main result of this section.

Theorem 3.4. If T is a tree with n vertices, then

Bn−1 = {d(M, X) |M ∈ V∗2 (T `, n− 1)}

is a reduced Groebner basis for In−1(T,X) with respect to the degree lexicographic order.

Proof. By Proposition 3 and Theorem 2.6 we only need to prove that S(f, g)→Bn−1 0 for all f, g ∈ Bn−1.

If M1,M2 ∈ V∗2 (T `, n− 1), then exists two paths P1 and P2 in T such that d(Mi, X) = d(T \ Pi, X).

We can suppose that either P1 or P2 is empty and that P1 6= P2, thus

S(d(M1, X), d(M2, X)) = xP c2 \P c1 d(T \ P1, X)− xP c1 \P c2 d(T \ P2, X),

where P ci = V (T ) \ V (Pi). If P1 ∩ P2 = ∅, then

S(d(M1, X), d(M2, X)) = xP1 d(T \ P1, X)− xP2 d(T \ P2, X).

By Lemma 3.2, S(d(M1, X), d(M2, X))→G 0.

If P1 ∩ P2 6= ∅, then this must be a path. If we set Q = P1 ∩ P2, then

S(d(M1, X), d(M2, X)) =xP1

xQd(T \ P1, X)− xP2

xQd(T \ P2, X),

and by Proposition 3.3 S(d(M1, X), d(M2, X))→G 0.

Next result gives us an alternative more compact description of minimal 2-matchings of T ` of size n−1.

Proposition 3.5. If Pu,v is the unique path on T that joins the vertices u and v, then

V∗2 (T `, n− 1) = {Pu,v ∪ {ww |w /∈ V (Pu,v)} |u and v are leaves of T}.

Proof. If P is any path on T , then by Proposition 1.9, M = P ∪ {ww |w 6∈ V (P )} is a minimal

2-matching of size n − 1 of T `. Therefore, we need to prove that if M ∈ V∗2 (T `, n − 1), then M =

Pu,v ∪ {ww |w /∈ V (Pu,v)} for some u, v leaves of T .

Let M ∈ V∗2 (T `, n − 1). If M has no edges, that is, M has n − 1 loops, then let u ∈ V (T ) such that

u 6∈ V (M) and v ∈ V (T ) such that uv ∈ E(G). Since M′ = {uv} ∪ (M \ {vv}) has size n − 1 and

that `(M′) ( `(M), then M is not minimal. Thus, M contains at least one path. Furthermore, since

T has n vertices, M has exactly a path. Let P =M\ `(M). If P ′ is a path on T such that P ( P ′,

then N = P ′ ∪ {uu|u 6∈ V (P ′)} is a 2-matching of size n− 1 and `(N ) ( `(M); a contradiction to the

minimality ofM. Therefore, P is maximal in the sense that P is equal to Pu,v for some leaves u, v and

M = Pu,v ∪ {ww |w 6∈ V (Pu,v)}.

Remark. If T is a tree with n vertices and m leaves, then Bn−1 contains(m2

)polynomials.

We finish this section with a conjecture about the minimality of the generating sets found in Theo-

rem 2.6. In the next section we present several examples of the validity of this conjecture.

Section 4 : Applications on critical group 63

Conjecture 3.6. If T is a tree and 1 ≤ j ≤ n, then

Bj = {d(M, X) |M ∈ V∗2 (T `, j)}

is a reduced Groebner basis for Ij(T,X) with respect to the degree lexicographic order.

We finish this section with an example of how to get the n− 1 critical ideal of a tree with n vertices.

Example 3.7. Let n1, n2, n3 ≥ 2 and J(n1, n2, n3) be the tree with a vertex r as root and three paths

Pn1 , Pn2 , and Pn3 from it of lengths n1, n2 and n3, see Figure III.3.

v1 v2 v3 v4 r v6 v7 v8

v9 v10

Figure III.3: The tree J(5, 4, 3).

If n = n1 + n2 + n3 − 2 = |J(n1, n2, n3)|, then by Theorem 3.4

In−1(J(n1, n2, n3), X) = 〈det(Pn1\ r,X),det(Pn2\ r,X),det(Pn3\ r,X)〉

In particular I9(J(5, 4, 3), X) = 〈x1x2x3x4 − x1x2 − x3x4 − x1x4 + 1, x6x7x8 − x6 − x8, x9x10 − 1〉.

4. Applications on critical group

Although the critical group of a tree is always trivial, critical ideals of trees can be used to obtain

information about the structure of critical groups associated to a large class of interesting graphs. This

section is devoted to present tree applications of the results of sections 2 and 3.

Trees of depth one and two

We begin with the trees of depth one or stars, which are the simplest trees, besides the path. Let S(m)

be the star with the root r and m leaves. If m ≥ 2, then is pretty clear that ν2(S(m)) = γ(S(m)) = 2.

Since |V (S(m))| = m+1, there are |V (S(m))|−2 = m−1 non-trivial critical ideals of S(m). Moreover,

if m ≥ 3 and 1 ≤ i ≤ m− 2, then is not difficult to see that

V∗2 (S(m)`, i+ 2) ={{vjsvjs}is=1 ∪ {vk1vr, vrvk2}

∣∣ j1 < · · · < ji and k1 < k2

}(III.1)

and therefore Ii+2(S(m), X) = 〈{∏is=1 xji

∣∣ j1 < · · · < ji}〉 for all i = 1, . . . ,m− 2.

Now, we continue with the trees with depth two. Let s ≥ 2 and T = T2(m1, . . . ,ms) be the tree of

depth two with r as the root and s branches with mi leaves each, see Figure III.4. Note that T2(∅)consist only of the root. If mi ≥ 2 for all 1 ≤ i ≤ s, then is not difficult to see that ν2(T ) = 2s. Since

n = |V (T )| = 1 + s+∑s

i=1mi, then T has n− 2s =∑s

i=1mi − s+ 1 non-trivial critical ideals.

In order to describe the non-trivial critical ideals we need characterize the minimal 2-matchings of T `.

Before to do this, we introduce some notation: Let v1, . . . , vs be the children of the root r of T , and Si

be the i branch of T , that is, the star induced by the vertices {vi, v(i,1), . . . , v(i,mi)}, see Figure III.4.

Also, for each 1 ≤ i ≤ s, let Vi be a subset of the leaves of Si of size at most mi − 2.


Proposition 4.1. If T = T2(m1, . . . ,ms) and M∈V∗2 (T `, 2s+ k) with k ≥ 1, then V (`(M)) is one of

the following sets:

1) U t (⊔si=1 Vi),

2) {r} t (⊔i∈I V (Si)) t (

⊔i/∈I Vi) for some I ⊆ {1, . . . , s},

where U is either empty or a leaf of T or two leaves of T with different father.

Proof. First we have two cases, when rr ∈ `(M) or rr 6∈ `(M). Suppose that rr ∈ `(M) and

let I = V (`(M)) ∩ {v1, . . . , vs}. The minimality of M implies that, if vi ∈ {v1, . . . , vs} \ I, then

|E(M) ∩ E(Si)| = 2. Thus |V (`(M)) ∩ V (Si)| ≤ mi − 2. Also, if i ∈ I since M is minimal, then

V (Si)⊆V (`(M)). Thus, if I = {i1, . . . , ij} and {v1, . . . , vs} \ I = {ij+1, . . . , is} with 0 ≤ j ≤ s, then

V (`(M)) = V (T2(mi1 , . . . ,mij )) t (⊔st=j+1 Vit).

Now, assume that rr 6∈ `(M). If r 6∈ V (E(M)), then M is a minimal 2-matching of⊔si=1 Si. Thus

M∩ Si is a minimal 2-matching of Si for each 1 ≤ i ≤ s and by III.1, V (`(M)) = U t (⊔si=1 Vi) with

U = ∅. Now, assume that r ∈ V (E(M)) and let J = {vi|rvi ∈ E(M)}. Since M is a 2-matching

|J | = 1, 2. By the minimality ofM we must have for each vi ∈ {v1, . . . , vs}\J that |E(M)∩E(Si)| = 2

and |V (`(M))∩V (Si)| ≤ mi − 2. Thus, V (`(M)) = U t (⊔si=1 Vi) with U a leaf if |J | = 1 or U two

leaves with different father if |J | = 2.

For the second part of the Proposition, if I ⊆ {v1, . . . , vs} for each vi 6∈ I let Ni be a 2-matching of Si

of size 2 and

MI = (⋃vi /∈I

Ni) ∪ (⋃vi∈I{vv | v ∈ Si}) ∪ {rr}.

Thus, by Proposition 1.9 MI is minimal 2-matching of T and V (`(M)) = V (T2(mi1 , . . . ,mij )) t(⊔st=j+1 Vit), where I = {vi1 , . . . , vij}. The case V (`(M)) = U t (

⊔si=1 Vi) is doing similarly.

Proposition 4.1 give us a complete description of the critical ideals of T = T2(m1, . . . ,ms). Moreover,

if M ∈ V2(T, 2s + k) and V (`(M)) = U t (⊔si=1 Vi), then k = |U | +

∑si=1 |Vi| and if V (`(M)) =

{r} t (⊔i∈I V (Si))t (

⊔i/∈I Vi), then k = 1 +

∑i∈I mi +

∑i/∈I |Vi| − |I|. Thus, if k = 1, then V (`(M) is

either a leaf of T or the root {r} of T and therefore

I2s+1(T2(m1, . . . ,ms), X) = 〈{xr} ∪ {xv | v is a leaf of T}〉.

In the next we give a description of the critical ideals of some trees of depth two with tree branches.

According to Proposition 4.1, the critical ideals of T2(m1,m2,m3) has two types of generators: monomi-

als and products of a monomial with the determinant of a tree of depth one. Thus, for each I ⊆ {1, 2, 3},let QI = det(L(ti∈ISi, X)). Also, let

P ir,s,t=

{r1∏l=1

x(1,il) ·s1∏l=1

x(2,jl) ·t1∏l=1

x(3,kl)

∣∣∣∣∣ 1≤ i1< · · ·<ir1≤m1, 1≤j1< · · ·<js1≤ m2,

1≤k1< · · ·<kt1≤m3, r1≤r, s1≤s, t1≤ t, r1 + s1 + t1 = i.

}

for all i, r, s, t ≥ 0. Moreover, by convention P ir,s,t = ∅ when either i, r, s or t is negative.


r

v1 v2 v3

v(1,1)v(1,2) v(1,m1) v(2,1)v(2,2) v(2,m2) v(3,1)v(3,2) v(3,m3)

Figure III.4: The tree T2(m1,m2,m3).

Example 4.2. Let T = T2(3, 4, 5) be the tree with tree branches, the first one with tree leaves, the

second one with four leaves and a third one with five leaves. Since n = |V (T )| = 16 and ν2(T ) = 6,

then T has 10 non-trivial critical ideals. Furthermore, by Theorem 2.6 and Proposition 4.1,

I6+i(T,X)=

〈xr · P i−11,2,3, P

i2,3,4, P

i−30,2,3 ·Q1, P

i−41,0,3 ·Q2, P

i−51,2,0 ·Q3, P

i−60,0,3 ·Q1,2, P

i−70,2,0 ·Q1,3〉 if 1 ≤ i ≤ 7,

〈P 82,3,4, P

50,2,3 ·Q1, P

41,0,3 ·Q2, P

31,2,0 ·Q3, P

20,0,3 ·Q1,2, P

10,2,0 ·Q1,3, Q2,3〉 if i = 8,

〈P 30,0,3 ·Q1,2, P

20,2,0 ·Q1,3, P

11,0,0 ·Q2,3〉 if i = 9.

Also, let T = T2(2, 2,m) be the tree of depth two with three branches, the first two with 2 leaves and

the third one with m leaves. Since n = |V (T )| = m + 8 and ν2(T ) = 6, then T has m + 2 non-trivial

critical ideals. By Theorem 2.6 and Proposition 4.1,

I6+i(T,X)=

〈xr · P i−10,0,i−1, P

i1,1,m−1, P

i−20,0,i−2 · {Q1, Q2}, P i−3

0,0,i−3 ·Q1,2〉 if 1 ≤ i ≤ m− 1,

〈Pm1,1,m−1, Pm−20,0,m−2 · {Q1, Q2}, Pm−3

0,0,m−3 ·Q1,2, Q3〉 if i = m,

〈Pm−20,0,m−2 ·Q1,2, P

11,1,0 ·Q3〉 if i = m+ 1,

Wired d-regular trees

A wired tree is a graph obtained from a tree by collapsing its leaves to a singular vertex. The critical

group of a wired regular tree (obtained from a regular tree) and some variants of them have been

recently study on [27, 36, 35].

For d ≥ 3, let T (d, h) be the d-regular tree of depth h and T ′(d, h) the tree obtained from T (d, h) by

delating one of his principal branches. In other words, T ′(d, h) is a tree of depth h in which the root

has degree d − 1, and all the other vertices, except the leaves, has degree d. We begin by calculating

the 2-matching numbers of T ′(d, h) and T (d, h).


Lemma 4.3. If h ≥ 2, then

ν2(T ′(d, h)) =

2

(d− 1)h+1 − (d− 1)

(d− 1)2 − 1if h is even,

2(d− 1)h+1 − 1

(d− 1)2 − 1if h is odd.

Proof. We will prove that if M is a maximum 2-matchings of T ′(d, h), then

|M| =

(d− 1) ν2(T ′(d, h− 1)) if h is even,

(d− 1) ν2(T ′(d, h− 1)) + 2 if h is odd.

(III.2)

Moreover, if r is the root of T (d, h) and h is odd, then degM(r) = 2, that is, r is saturated.

Since |M| = ν2(T ′(d, h)), the relations above leads to the fact that ν2(T (d, h+2)) = (d−1)2ν2(T (d, h))+

2(d− 1) for even h. This difference equations and the initial value ν2(T (d, 2)) = 2(d− 1) leads to the

formulae for even h. The formulae for odd h is a consequence of the even case.

To prove III.2 we use induction on h. The base step, i.e. T (d, 2) and T (d, 3) can be done by inspection.

Suppose that M is a maximum 2-matchings of T ′(d, h+ 1). Note that T ′(d, h+ 1) has d− 1 copies of

T ′(d, h) so |M| ≥ (d− 1)ν2(T ′(d, h)).

Suppose that h+ 1 is even. IfM has just one edge on the root of T ′(d, h+ 1) then, let w be the vertex

connected to r by an edge of M. Let T ′w(d, h) be the copy of T ′(d, h) inside T ′(d, h+ 1) rooted at w.

As Mw = M∩ T ′w(d, h) has degree at most one on w and h is odd, by the induction hypothesis it

follows that Mw is not maximum on T ′w(d, h). Thus,

|M| < ν2(T ′(d, h)) + 1 + (d− 2)ν2(T ′(d, h)) = (d− 1) ν2(T ′(d, h)) + 1,

which implies that |M| = (d− 1)ν2(T ′(d, h)). The case in which M has degree 2 on r can be treated

in the same way.

Suppose that h + 1 be odd. As T ′(d, h + 1) \ r consist of d − 1 copies of T ′(d, h) it follows that

ν2(T ′(d, h + 1)) ≥ (d − 1)ν2(T ′(d, h)) + 2. Even more, in each one of this copies there exist a 2-

matchings of size ν2(T ′(d, h)) with degree 0 on their root. The union of this 2-matchings is a 2-

matching of T ′(d, h + 1) with degree 0 on r and on their pendants vertices. Thus, ν2(T ′(d, h + 1)) ≥(d− 1)ν2(T ′(d, h)) + 2 and M must be composed of 2 edges at r and a maximum 2-matching in each

copy of T ′(d, h).

Corollary 4.4. If h ≥ 3, then

ν2(T (d, h)) = 2(d− 1)h − 1

d− 2

Proof. Since ν2(T (d, h)) = ν2(T ′(d, h)) + ν2(T ′(d, h− 1)), the result it follows from Lemma 4.3.

The proof of Lemma 4.3 can be refined to describe the first non trivial critical ideal of T ′(d, h) and

T (d, h). Since the proofs for T ′(d, h) and T (d, h) are very similar, we only present the proof for T ′(d, h).


Corollary 4.5. If h ≥ 3 and d ≥ 4, then

Ik(T′(d, h), X) =

⟨{xv

∣∣ v is a leaf of T ′(d, h)}, Ik′(T ′r(d, h− 2), X)⟩,

where k = ν2(T ′(d, h)) + 1 and k′ = ν2(T ′(d, h− 2)) + 1.

Proof. Each minimal 2-matching of T (d, h)` of size ν2(T ′(d, h))+1 consist of a maximum 2-matchingMof T (d, h) and a loop on one of the vertices not covered byM. Let j = 1 when h is odd and j = 0 when

h is even. From the inductive description of the maximum 2-matching given in the proof of Lemma 4.3

it follows that the leaves of T ′r(d, h − 1), . . . , T ′r(j + 1) are saturated. Then, if M ∈ V∗2 (T ′(d, h)`, k)

the loop can only been on the leaves of T ′(d, h), T ′r(d, h − 2), . . . , T ′r(d, j). Furthermore, each leaf of

T ′(d, h), T ′r(d, h− 2), . . . , T ′r(d, j) is free for some maximum 2-matching of T ′(d, h).

Corollary 4.6. If h ≥ 3 and d ≥ 4, then

Ik(T (d, h), X) =⟨{xv

∣∣ v is a leaf of T (d, h)}, Ik′(Tr(d, h− 2), X)⟩,

where k = ν2(T (d, h)) + 1 and k′ = ν2(T (d, h− 2)) + 1.

Let T (d, h) be the wired tree obtained from T (d, h). In [36], Toumparaky calculate the rank, the

exponent and the order of the critical group K(T (d, h)) of T (d, h). Is not difficult to see that T (d, h)

is equal to the multigraph obtained from adding a new vertex v to T (d, h− 1) and d− 1 edges between

each leaf of T (d, h−1) and v. Thus, using Corollary 4.5 we can calculate easily the rank of K(T (d, h)).

Proposition 4.7. If d ≥ 4, then the critical group of T (d, h) has rank (d− 1)h and its first non trivial

invariant factor is equal to d.

Proof. Let n = |T (d, h − 1)| and f1, . . . , fn be the invariant factors of K(T (d, h)). By [20, Propo-

sition 3.7], for each i = 1, . . . , n, f1 · · · fi can be determined by evaluating the set of generators for

Ii(T (d, h), X) and after that calculating its greatest common divisor. Since k = ν2(T (d, h− 1)), clearly

f1 = · · · = fk = 1 and K(T (d, h − 1) has rank n − k = (d − 1)h. Furthermore, since degT ′(d,h)(v) = d

for any leave v of T ′(d, h), then using Corollary 4.5 and induction on h we get that fk+1 = d.

In [27], Levine describe the critical group of the multigraph T h obtained from T ′(d, h) by collapsing

all the leaves to a single vertex v and adding an edge between v and the root of T ′(d, h), which is very

similar to T ′(d, h). Note that, at difference to T (d, h), the multigraph T h is d-regular. In a similar way

that in Proposition 4.7, we can use Corollary 4.5 to calculate the rank of K(T h) by evaluating the first

non trivial critical ideal of T ′(d, h− 1).

Proposition 4.8. If d ≥ 4, then the critical group of T h has rank (d− 1)h. Furthermore, the first non

trivial invariant factor is equal to d.

Proof. Let v the vertex of T h and T ′(d, h − 1) obtained by collapsing the leaves of T ′(d, h). Since

T h\v = T ′(d, h−1)\v, the result it following similar arguments that those given in Proposition 4.7.


Unfortunately, the technic described by Levine in [27] can not be used to calculate the critical group

of T (d, h). This technique consist on use a recursive relation among the critical group of T h and the

critical groups of his principal branches [27, Proposition 4.3]. We found certain similarity between

this recursive relation and the description of the first critical ideal of T ′(d, h) given in Proposition 4.5.

Thus, we not doubt that a deeper study of the critical ideals of T ′(d, h) allow us to replicate all the

results obtained by Levine [27] and Toumpakary [36].

Arithmetical trees

An arithmetical graph is a triplet (G,d, r) given by a graph G and d, r ∈ Z|V (G)|+ such that (Diag(d)−

A)r = 0, where A is the adjacency matrix of G. Any graph G belongs to an arithmetical graph in a

natural way, just taking d as its degree vector and r = (1, . . . , 1)t. The matrix M = Diag(d)−A arise

in algebraic geometry as an intersection matrix of degenerating curves, see [29, 30] and the references

contained there for more details.

Given an arithmetical graph (G,d, r), we define its critical group K(G,d, r) (also called the group

of components) as the torsion part of Z|V (G)|/Im(M). In [29], Lorenzini proved that the Z-rank of

K(G,d, r) is equal to n − 1. Furthermore, if the Smith Normal Form of M is diag(f1, . . . , fn−1, 0),

then K(G,d, r) = Zf1 ⊕ · · · ⊕ Zfn−1 . Since for each 1 ≤ j ≤ n − 1,∏ji=1 fi is the greatest common

divisor of the j-minors of M and M = L(G,d), it follows that 〈∏ji=1 fi〉 is the generator of Ij(G,d),

the j-critical ideal of G evaluated in d.

Thus, the invariant factors of K(G,d, r) can be found as follows: First, find a set of generators of the

critical ideals of G. After that, we evaluate them on d and compute its greatest common divisor. For

instance, consider the family of arithmetical graphs associated to the reduction of elliptic curves of

Kodaira type I∗n. For any m ∈ N, let C5,m the tree obtained by identifying the center of a star with

two leaves with each leaf of the path Pm+1, see Figure III.5.

1

2

3

4

5 67 81

2

3

4

5 67 8

1

2

3

4

5 6

7 8

(a) (b) (c)

Figure III.5: The tree C5,m and the two types of 2-matchings of size m+ 3.

Now, we will describe the critical ideals of C5,m. First, since V (C5,m)\{v1, v3} induces a path isomorphic

to Pm+3, it follows that ν2(C5,m) ≥ m+ 2. Moreover, is not difficult to check that ν2(C5,m) = m+ 2.

Thus, by Theorem 2.8, γ(C5,m) = m + 2 and C5,m has only 3 non trivial critical ideals. The m + 5-

critical ideal is the determinant of the generalized Laplacian matrix. For simplicity, we will assume

that m ≥ 5. By Proposition 3.5 we get that

Im+4(C5,m, X) = 〈x1x3, x1x4, x2x3, x2x4,P1,2P7,8 − x1x2P9,8,P3,4P7,8 − x3x4P7,10〉,


where Pi,j = det(P (vi, vj)) and P (vi, vj) is the unique path in C5,m that join the vertices vi and vj . Note

that, det(C5,m \P (v3, v4), X) = P1,2P7,8−x1x2P9,8 and similarly in the case of det(C5,m \P (v1, v2), X).

Finally, in Figure III.5 are sketched the two types of minimal 2-matchings of C5,m of size m+ 3. Thus

Im+3(C5,m, X) = 〈x1, x2, x3, x4,P7,8〉.

Now, taking d5,m = (2, . . . , 2) and r5,m = (1, 1, 1, 1, 2, . . . , 2)t we get that (C5,m,d5,m, r5,m) is an

arithmetical graph. Since γ(C5,m) = m + 2, fi = 1 for all 1 ≤ i ≤ m + 2. On the other hand,

using [20, Corollary 4.5] we get that the polynomial Pi,j evaluated in d = (2, . . . , 2) is odd if and

only if the path P (vi, vj) has an even number of vertices and P1,2 and P3,4 evaluated in (2, 2, 2) are

equal to 4. Thus, fm+3 = 1 when m is odd, fm+3 = 2 when m is even. Finally, since fm+3fm+4 =

Im+4(C5,m, (2, . . . , 2)) = 4, then

K((C5,m,d5,m, r5,m)

)=

Z22 if m is even,

Z4 if m is odd.

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75

Index

ck( ), see cone of a graph

Cn( , ), 6

d( , X), 55

dG( ), 28

G( ), see duplication of a graph

G`, 53

K(M), 5

K( ), 2

Kn( , ), 6

KM,M , 15

KM,m, 15

Km,m, 15

L(G,X), 25

L(G,XG), 25

L(G, d), 28

LM,M , 15

LM,m, 15

Lm,m, 15

M(I; J), 1

minorsi(M), 1

M [I; J ], 1

MM,M , 15

MM,m, 15

Mm,m, 15

pG(t), 34

Pn( , ), 6

S(m), 63

Tn, 28

Tn( , ), 6

Ts(. . .), 63

XG, 25

∆i(M), 5

Φm( , ), 14

α( ), 30

`( ), 53

γ( ), 30

γP( ), 30

V1(G), 37

V1(S), 57

V2( ), 50

V∗2 ( ), 54

ω( ), 30

2-matching, 50

minimal, 53

adjacency matrix, 2

arithmetical graph, 68

Buchberger criterion, 3

characteristic polynomial, 34

clique number, 30

cokernel, 1

cone of a graph, 11

critical

group, 2

ideals, 26

module, 5

76

degree lexicographic order, see monomial order

degree vector, 28

duplication of a graph, 11

free part, 1

generalized Laplacian, 25

Grobner basis, 2

reduced, 2

i-critical ideal, 26

invariant factors, 2

Laplacian matrix, 2

reduced, 2

leading

coefficient, 2

power, 2

term, 2

minor, 1

monomial order, 2

degree lexicographic, 3

reduced Grobner basis, see Grobner basis

reduced Laplacian, see Laplacian matrix

reduction, see strong reduction

S-polynomial, 2

Smith Normal Form, 1

stability number, 30

star, 63

strong reduction, 3

torsion part, 1

trivial graph, 28

wired tree, 65

critical ideals of graphs - cinvestav · g], the determinantal ideals of l(g;x g) are ideals on z[x...

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