ROOTS OF CHROMATIC AND INDEPENDENCE

POLWOMIALS

BY

Car1 Andrew Hickman

SUBMITI'ED IN PARTIAL FULFILLMENT OF THE

REQUlREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPWY

AT DALHOUSIE UNTVERSITY

HALIFAX, NOVA SCOTlA

JUNE 7,2001

@ Copwght by Car1 Andrew Hickman, 2001

Acquisitions and AcquiSitins et û i i i m p h i c Senrices seniices bibiiographiques

The author has gmted a non- L'auteur a accordé une licence non exclusive b c e dowing the exclusive permettant à la National Li- of Cana& to Biblioth&pe nationaie du Canada de reproduce, loan, distniute or seii reproduire, prêter, âistribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous papa or electrorîic formats. la forme de microfiche/^ de

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The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts from it Ni la thèse ni des extraits substantiels may be p ~ t e d or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.

In loving memory of Allan Hzckman.

Contents

Abstract vii

Acknowledgment

1 Introduction and Background

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 The Riemann Sphere . . . . . . . . . . . . . . . . . . . . . . .

1.3 An Ovemiew of the Thesis . . . . . . . . . . . . . . . . . . . . . . . .

2 Roots of Chromatic Polynomials

2.1 Generalized Theta Graphs . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Theta Graphs with 5 8 Paths . . . . . . . . . . . . . . . . . . 2.1.2 Theta Graphs with 3 Paths: An Explicit Farnily having Roots

with Negative Real Part . . . . . . . . . . . . . . . . . . . . . 2.2 Large Subdivisions of Gr+ . . . . . . . . . . . . . . . . . . . . . .

2.2.1 Recursive Fiimilies of Polynomials . . . . . . . . . . . . . . . . 2.2.2 An Expression for the Chromatic Polynomials of Subdivisions

2.2.3 Uniform Subdivisions and the Limits of their Chromatic Roots

2.2.4 Application 1: Chromatic Roots with Negative Real Part . . .

2.2.5 Application 2: Bounding the Chromatic Roots of Large Subdi-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . visions 39

Roots of Independence Polynomials 43

3.1 Location of Independence Roots of some Families of Graphs . . . . . 46

. . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Weii Covered Graphs 46

. . . . . . . . . . . . . . . . . . . . . . 3.1.2 Comparability Graphs 49

. . . . . . . . . . . . . . . . . . 3.1.3 On Further Classes of Graphs 53

. . . . . . . . . . . . . . . . 3.2 The Independence Attractor of a Graph 54

3.2.1 Julia Sets and the Iteration of Rational Functions . . . . . . . 55

3.2.2 Independence Attractors of Graphs: a General Theory . . . . 61

. . . . . . . . . . . . . 3.2.3 Graphs With Independence Number 2 67

. . . . . . . . . . . . . . . . . 3.2.4 Beyond Independence Number 2 73

4 Open Problems and Future Directions 86

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Chromatic Roots 86

. . . . . . . . 4.1.1 Bounding Chromatic Roots in terms of Corank 87

. . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Inde pendence At tractors 88

. . . . . . . . . . . . . 4.2.1 Independence Attractors of Complexes 89

A A Maple Procedure for Independence Attractors 95

Bibliography 98

Abstract

Two polynomials arise naturally h m the notion of an independent set of vertices

in a graph G: (i) the chromatic polynomid, *(G, x) = x,,, - r k x ( ~ ) , where r k is the

number of partitions of V ( G ) into k independent sets (and x(" a fdling factorial);

and (ii) the independence polynomial i c (x ) = &, - ik xk, where ik is the number of

independent sets in V of cardinality k. We study here the roots of these polynornials.

each respect to a specific graph operation: chromatic roots with respect to edge

subdivision, independence roots with respect to graph composition.

For a connected graph G of CO-rank k = IEl - IV1 + 1, it is known that any

chromatic root z satisfies (t - II 5 k. We prove that large subdivisions of G. while

not changing its c o - r d , draw the chromatic roots close to the disk Ir - 11 5 1.

For 'uniform' subdivisions, we describe the iimit points of the roots, and in t u m

characterize graphs having a subdivision with a chromatic root with negative real

part. In k t , infinitely many such roots are achievable fiom graphs of corank 2, the

3-as, theta graphs. And for each 3 5 k < 8, the k-ary theta graph with path lengths

2 has a chromatic root r which maximizes lz - 11. Independence polynomials are (essentially) closed under graph composition. We

prove this, and apply it to families of well covered and comparability graphs, hding

the topological closures of both real and cornplex independence roots. For higher

composites of a graph G with itself, we prove that their independence roots con-

verge (in the Hausdorff topology) to the Julia set of io(x) - 1, thereby associating a

kactal with G. For graphs with independence number 2, we determine when these

Fractals are connected. Further, the join of all sufEciently many copies of any graph

vii

has a disconnected fkactal, proving the existence of many connected graphs with a

disconnected independence /ractal

Acknowledgment

I am sincerely gratehil to my supervisor, Jason Brown, for his genuine interest in

my research and career, for never being 'too busy' to set some time aside for getting

together. for stimulating conversations and valuable advice on many topics. for his

patience and for making s u e that 1 stayed on track when the crunch was on. He is

everything that one could hope for in a supervisor. and more.

To my parents Car1 and Agnes, my brother Greg, my grandmother Jessie and my

beautiful fiancée Mona. thanks so much for al1 of your love, kindness. encouragement

and support.

1 want to thank Dr. Richard Xowakowski for his insight and ve- helpful discus-

sions on a recent career decision 1 was presented with.

To Dale Garraway, I am grateful for the countless hours we spent toget her prepar-

ing for those dreaded comprehensive exams.

And many thanks indeed to both Dalhousie University and the Naturd Sciences

and Engineering Research Council of Canada for financial support during my years

as a graduate student .

Chapter 1

Introduction and Background

1.1 Introduction

One method of investigating a combinatorial sequence is to associate a generating

function with the sequence, and study the fùnction itself, including its roots (or 'ze-

ro~') . In this thesis, we will study the roots of two generating functions - chromatic

and independence polynomials - which aise naturdy boom the notion of an indepen-

dent set of vertices in a gaph . In section 1.3, we will lay out the plan for the thesis.

For now, we sirnply introduce the two polynomials.

The chromatic polynomzal of a g a p h G(V, E) is the function r (G, x) = x,, - , rcdk) ,

where r k is the number of ways of partitionhg V into k independent sets, and x(') is

the falling factorid x(x - 1) - (z - k + 1). For x a positive integer, r (G, x) counts

the number of proper vertex colourings of G using at most x given colours. Expand-

hg, then collecting terms of like degree, leads to an expansion of the f o m *(C, x) =

~ k > O ( - î ) k ~ ~ n - k , where n = IVI. As it happens, the sequence (bo,bl , . . . , b n - l ) is

also combinatorial: bk counts the number of broken circuits (cf. [IO]) in G.

Initial interest ([i2, 13, 141) in the mots of chromatic polynomials stemmed from

the former four colour conjecture, which states that 4 cannot be a chromatic root of

any planar graph. Roots of chromatic polynomials have since emerged as a rather

fascinating topic in its own right , having attracted considerable attention (cf. [9, 11,

18, 21, 22, 45, 57, 601). Chromatic polynornials also arise in statistical physics as

zero-temperature limits of the partition function of the Potts mode1 antûerr~rnag~et

on G (cf. [55]). Its roots are closely h k e d to phase transitions (611, and as such have

received interest from physicists (52, 53, 541 in addition to mathematicians.

The independence polynomial of G(V, E ) is the function ic (x) = x,,, - G xk, where

ik is the number of independent sets in V having cardinality k. It is the generating

function for the 'independence vector' (io, il, . . . , ig), and generalizes the more familiar

matchzng polynomial (cf. [39]), M(G, x) = &,(-l)kmkP-2k, where rnk is the

number of matchings in G on k edges. Indeed. a rnatching in G is an independent

set in L(G) (the line graph of G), and mere inspection reveals that M(G. x) = xn .

~ L ( G ) (-x-*).

Matching polynomials also arise in stat ist ical physics, as monomer-dimer partit ion

functions (cf. [42]). Again, the roots are associated with phase transitions. The

roots of matching polynomials are al1 real (cf. 1421). Whiie the sarne is not true of

independence polynomials, it is known ([35]) that at least one their roots is real: in

fact, a root of smallest modulus is necessarily real. Further results on independence

polynomials and their roots can be found in [23, 35. 40, 421.

1.2 Background

Some basic terminology and results will be needed for the chaptea which follow.

1.2.1 Graphs

A gaph G(V, E ) consists of a finite, nonempty set V (the vertices of G) and a multiset

E of o n e and tweelement subsets of V (the edges of G). If not d of the elements of

E are distinct, then G is said to have parullel edges, since there are pairs of vertices in

G with at least two edges between them. A graph with no pardei edges is a simple

graph. Those edges in E which are one-element subsets of V are called loops.

Two vertices u and v in G are adjacent if they are joined by an edge, that is, if

{a, v) E E. We often write UV instead of {a, v) . The vertices u and v are the ends

of e = UV, and e is incident with both u and v; both IL and v are incident with e.

The degree of a vertex v E V, written degv, is the number of edges in G which are

incident with v. The maximum degree of a vertex in G is denoted by A(G). If every

vertex in G has degree k; then G is k-regular.

A graph H is a subgraph of G if V(H) E V(G) and E(H) C_ E(G). If H is a

subgraph of G such that for ail pairs of vertices u and v in V ( H ) it is true that

UV E E ( H ) if and only if UV E(G), t hen H is an znduced subgraph of G.

A cycle of length k 2 2 in G is a finite sequence voelvle2 . . . etvo whose terms are

alternately vertices and edges, with e, = vi-lvi for i = 1,. . . , k, and such that no

vertex is repeated. The gzrth of G is the lengt h of the shortest cycle in G (or oa if G

has no cycles). If G has n vertices and consists of only a cycle, then G is an n-cycle.

written G = C,. Removing any single edge from an n-cycle leaves a path P, on n

vertices; P, has n - 1 edges (the length of the path).

If G and H are simple graphs such t hat V ( G ) = V ( H ) = V o and, for al1 distinct u

and .u in V , UV 6 E(G) if and only if uv E E(H) , then H (resp., G) is the complement - - of G (resp., H). We write H = G (or G = H ). The union G = Cl U G2 of two

graphs G1 (VI, E l ) and G2(b1 E2) is the graph with vertices V(G) = V(GJ U V ( G 2 )

and edges E(G) = E(GI) U E(G2). I f , in addition, VI n V2 = 0' t hen G is the disjoint

union of GI and G2, written G = Gi u G2. The simple gaph G on n vertices having an edge between every pair vertices is

the complete graph K,. Its complernent is the empty graph of order n, written En. For a graph G(V, E), a subset U of V is independent if, for a l l u and v in II,

edge UV is not in E. The largest size of an independent set in G is the independence

number of G. A k-partite gaph is a graph whose vertex set can be partitioned into

k independent subsets. The graph is called bipartite if k = 2. If G is a k-partite

graph, and IIl, . . . , CI' is a partition of V(G) into k independent subsets of cardinality

nl , . . . , nh, respectively, and E (G) consists exactly of ail possible edges between the

sets IIl, . . . , IIk, then G is complete k-partite, and we mite G = K, ,,---, n,. It is not

- hard to see that K, ,,...,,, = K,, + - * - +Zn&. If k = 2, then G is complete bipartite.

A graph is connected if there exists a path between any two of its vertices, and

is disconnected otherwise. The components of a graph are its maximal connected

su bgrap hs.

The connectzvity of G, written n(G), is the minimum number of vertices whose

removal disconnects G or reduces it to Ki. G is said to be k-connected if n(G) 2 k.

A cut set in G is a subset of V(G) whose removal increases the number of components

in G. If v E V(G) and {v} is a cutset in G, then ,u is called a eut vertex. So a graph

with cut vertices is not 2-connected. A block in G is a subgraph that is maximal

subject to being either K2 or 2-connected. An edge e of G whose removal increases

the number of componenets of G is called a bridge.

A colouring of a graph G is an assignment of colours to its vertices in such a way

that no two adjacent vertices get the same colour. If (at most) k colours are used.

we have a k-coloufing of G, and G is k-colourable. More precisely a k-colouring of

C is a function f : V ( G ) + (1,. . . ! k} such that, for all u and u in V ( G ) , we have

that UV E E (G) implies f (u j + f (v). The chromatic number, x(G), is the minimum

number of colours required to colour G. Any colouring of G partitions V(G) into

independent subsets? which we cal1 colour classes, one class for each colour use&

1.2.2 Roots of Polynomials

Throughout the thesis, we wiil make use of fundamental results on the mots (or

zeros) of (univariate) polynomials with real and complex coefficients. We collect

these results here, beginning with a necessary condition for a rational number to be

a root of an integer polynomial. The notation k[x] denotes the ring of polynomials

in x with coefficients from k.

Theorem 1.2.1 (Rational Root Theorem) k t f (z) = @zi E Z[x] with

# O. If a/b E Q is a root of f(x), with gcd(a, b) = 1, then al% and blta,,.

A polynomial f with real coefficients is Hunuitz quusi-stable [5, 581 if every root of

f has nonpositive real part. The statement of the Hermite-Biehler Theorem proved in

Gantmacher [37] is ac tudy a criterion for deciding whether every root of a real poly-

nomial has strictly negative real part. Wagner [58] deduced Erom this an analogous

criterion for Hurwitz quasi-stability. It is the latter which we shall c d the Hermite-

Biehler Theorem. As in (581, a polynomial is standard if it is either identicdy zero

or has positive leading coefficient, and is said to have on$ nonpositive rems i f it is

either identicdy zero or has all of its roots real and nonpositive.

Theorem 1.2.2 (Hermite-Biehler) Let P ( x ) E W [z] be standard, and w i t e P(a) =

Pe(x2) + xP,(x2) . Set t = x2. Then P ( x ) is Humitr quasi-stable if and only if both

PJ t ) and PJt) are standard, have only nonpositiuc rems, and P&) 4 Pe(t) .

Roughly speaking (and made precise in [SI), the notation Po(t) i Pe(t) says t hat

the roots of P,(t) 'interlace' the roots of P,(t). but we need not concern ourseives

with that here. In fact, we shdl use only the following.

Corollary 1.2.3 If either P,(t) or Po@) has a nonreal mot (and 2s not identicdiy

zero), then P ( x ) is not Hunvitz quasi-stable.

Now Sturm's Theorem (cf. [46]) gives rise to a useful test for deciding whether a

red polynomid has a nonred root. We Say that two consecutive terms of a sequence

s = (Q, al , . . . , ak) of nonzero real numbers have a sign variation if they have opposite

signs, and denote by Var s the number of sign variations of S. If s contains zero entries,

then Vars is definecl to be the number of sign variations of the subsequence of nonzero

temw of S.

The Stuma sequence of a real polynomial f ( t ) of positive degree is fo, f i , fi, . . . , where fo = f , fl = f', and, for i 2 2, fi = -rem(fi-i, fi-?), where rem(g, h) denotes

the remainder upon dividing g by h. The sequence is terminated at the last nonzero

fil which is easily seen to be a constant times the greatest common divisor of f and

f' (just compare the process to the Euclidean Algonthm).

Theorem 1.2.4 (Sturm's Theorem) Let f ( t ) E R[t] have positive degree, und

suppose ( fo, fi, . . . , fk) zs its S t u m sequence. Let a < b be reais that are not mots

of f Then the number of distinct roots of f in (a, 6 ) zs V ( a ) - V ( b ) , where V(c)

var(fo(c), fi(c), - fk(c))-

A proof can be found in [46]. Now let us Say that the Sturm sequence (fa, fi, . . . , fk) of f (t) has gaps in degree if there is a j 5 k such that deg fj < deg f,-i - 1. If there is

a j 5 k such that fj has negative leading coefficient, then we Say the Sturm sequence

contains a negative leading coeflcient.

We shall make important use of the following corollary to Sturm's Theorem which

is not explicitly found in the literature (a similar statement, though stated inconectly,

is found in [4, p. 1761 ) .

Corollary 1.2.5 Let f ( t ) be a real polynornial whose degree and leadzng coeficient

an! positive. Then f (t) h a al1 real mots if and on$ zf its Stunn sequence has no gaps

in degrile and no negative leading coeficients.

Proof Let us begin with a few observations. We write f = g - h, where g = gcd( f , f ') . Then the number of distinct roots of f is exactly deg h. a s the roots of g are the

multiple roots of f . Consider the Sturm sequence Sf(t) = ( fo, f l , . . . , fk) of f . Recall

that fo = f and fk is a (nonzero) constant times g. Then since deg f = degg +deg h,

we have that the number of terms in S f ( t ) is k + 1 5 deg h + 1, with equality exactly

when it has no gaps in degree.

We d e h e V(-oc) and V(m) to be, respectively, V(-M) and V ( M ) , where M > O

is any number large enough that ail real roots of each fi (i = O, 1,. . . , k) lie in

(-M, M). It is clear that

V(-KI) = Var ((- l)dwfo Icoeff fo, (-l)dqfllcoeff fi, . . . , (- ~)~@%x~eff fk)

and

V(m) = Var (Icoeff fo, lcoeff fi, . . . , lcoeff fk),

where lcoeff ~ denotes the leading coefficient of $.

Al1 real roots of f lie in (-M, M) as f = fo. Then Sturm's Theorem says that

the number of distinct real roots of f is V(-cm) - V(w).

With these observations, we prove the result. If Sf(t) has gaps in degree, then it

has k + 1 < deg h + 1 terms, so in particular V(-CIO) 5 k < deg h , and so

# distinct real roots of f = V(-oo) - V ( w )

s V(-W)

< degh

= # distinct roots of f ,

which implies that f has a nonreal root. If Sf(t) has no gaps in degree but has a

negative leading coefficient. then let j be the first i such that lcoeff f, < O. Then

lcoeff fj-i > O (as lcoeff fo is), and since deg f, = deg fj-l - 1, we have that

(-l)degf~-llcoeff fj-, and (-l)dghlcoeff fj have the same sign, so V(-oo) < k =

deg h, and again f has a nonreal root.

Conversely, if Sf(t) has no gaps in degree and no negative leading coefficients.

then k = deg h, V(-w) = k, and V ( w ) = O, so

# distinct real roots of f = V(-os) - V(m)

= deg h

= # distinct roots of f ,

which says f has all real roots.

For c any positive real number, it is easy to see that if, on obtaining the term

fi (O 5 j 5 k) in the construction of the Sturm sequence (fo, f i , . . . , fk) of f ( t ) , we

were to change fj to c fj before continuing, then the resulting sequence would d8er

hom ( fo, fi, . . . , fk) only in that some fi's would now become ch, and so clearly that

sequence could be used in place of (fo, fi,. . . , fk) when applying Theorem 1.2.4 or

Corollary 1.2.5. In fact, we may perfonn multiplications like this at any number

of steps (by repeatedly applying the above argument), and we consider any to be a

Sturm sequence of f (t). We shall make use of this observation in Section 2.1.2.

1.2.3 The Riemann Sphere

In Chapter 3, we will undertake a study of fractals which mise in a nat ural way from

independence polynomials of gaphs. In the process, we will make nontrivial use of

results hom the theory of iterating rational functions, an understanding of which

requires knowledge of a few specific facts about metric spaces - the Riemann sphere?

in particular. This material, which we now review, c m be found in the works of

Barnsley [6] and Beardon (71.

We assume the reader is f a d i a r with the definition of a metric space (X, d): a

complete metric space, and a compact subset of a metric space. The collection R(X) of compact subsets of a metric space (X, d) forms a metric space in itself: if A and B

are two compact subsets of (X, d), then the distance d(A, B) [rom A to B is given by

d ( A , B ) = maxmind(a, b). aEA bEB

The definition is not symmetric. However, if we define

then hd is a metric on H ( X ) ; in fact, if ( X , d ) is a complete metric space, then

(R(X), hd) is a complete metric space.

On the space X = C of complex numbers, the Euclidean rnetric 1 - 1 measures

the distance between points z and w as Ir - wl; it is well known that (C, 1 1) is a complete metric space. As is the case with (R,1 l), however, the notion of a

sequence 'converging to i . t y ' requires a separate definition. While the fractals we

wiU consider in chapter 3 are bounded in (@, 1 1) , their study involves iteration theory

of polynomials, where it is important to eliminate this special s i ~ ~ c e of inhity.

First, an abstract point, denoted by oo, iî adjoined to @, forming the set @, =

C U {a), called the eztended complex plane, or Riemann sphem. To obtain a metric

on @,, we constmct a mode1 of @, as a sphere, as follows. IdentifJr C with the

horizontal plane in IR3 containing the origin, and let S be the sphere in R3 with unit

radius and centered at the origin. The stereagruphzc projection of C into S is the

map r which takes a cornplex number r and projects it Linearly towards the top point

(O,& 1) of S until it meets S at a point n ( z ) ; also, we set ~ ( o o ) = (0,0,1). This leads to two natural metrics on C,: the chordul and sphericul metrics o and 00,

respectively, are defineci as follows. If z and w are two points in C,, then

cr(z, w) = the Euclidean distance (in IR3) between ~ ( z ) and r(w);

~ ( z , w) = the great circle distance (on S) between ~ ( z ) and *(ut).

Recall that two metrics d l and d2 on X are equivalent if there exist positive real

constants ci and c2 such that

for al1 x, y E X. The chordal and spherical metrics on C, tm out to be equivalent.

Further, on a bounded subset of (ê, 1 0, each is equivalent to the Euclidean metric,

Theorem 1.2.6 Let % be a bounded subset of (@, 1 - 1). Then, on X, the Euclideun,

spherical and chorda1 rnetrics are equivalent.

Proof It suffices to show that 1 - 1 is equivalent to o on X. As % is bounded with - respect to 1 1, there srists a number M > O such that X is containeci in an M- radius

212 - ul of O. For any two points z and Y in X, then, u(z, w ) =

(1 + I~l)l'~(l + Iwl)l'* (a well

known explicit formula (cf. [7]) for a(z, ut)) satisfies

and 1 - 1 is thus equident to a on X, completing the proof.

From this, we may conclude:

Theorem 1.2.7 Let X be a compact subset of (C, 1 - 1) , and {A , } a sequence in X(X).

Let A E ?i(W). Then A, + A zuith respect to h, (or h,) if and only if A,, -t A with

respect to hl., .

In chaper 3, we consider sequences {A,) which aise Erom root sets of an iterated

graph-theoretic polynomial, and which are always bounded in (@, 1 1). We are natu-

r d y led to ronsider their limit in that space. Theorem 1.2.7 tells us that their limit

in (Cm, q) is identical, which then enables us to make important use of iteration

theory in describing that limit.

1.3 An Overview of the Thesis

The roots of a graph polynomial are studied h m various standpoints, from bounding

the roots, to determining accumulation points for the roots, to finding regions in the

complex plane or real line containing no root at dl. The effects that familiax graph-

theoretic operations have on the roots are of interest as well, and this will be a

common thread in the thesis.

Our study of roots of chromatic polynomials ('chromatic roots') in chapter 2 is

directly related to the operation of subdzvidzng an edge of Go that is, replacing an edge

with a path. If G consists of nothing more than two vertices joined by one or more

edges, then subdividing edges in G gives rise to what are known as generalized theta

gmphs. Numerical calculations suggest that among all generalized theta gaphs on k

paths, the one with a chromatic root z that maiamizes Iz - II is the graph with all

path lengths wual to 2; we will prove that for k 5 8, this is indeed the case. Roots

of chromatic polynomials lying in the left-haIf plane have &O received attention, and

we will prove that infinitely many such are achievable among theta graphs on 3 paths,

t hereby providing a family of minimal CU-d (cf. chapter 2) having chromatic roots

with negative real part. Moreover, a graph G will possess a subdivision having a

chromatic root with negative real part if and only if G has a theta-subgraph. At the

same tirne, however, large subdivisions of all the edges in G draw the roots close to

the disk lz - il 5 1. In chapter 3, we tum o u attention to roots of independence polynomials ('inde-

pendence roots'). The work is based largely on the lexicographzc product (or graph

composition) operation, which involves 'replacing' each vertex of G with a graph H.

We begin the chapter with the observation that independence polynomials are essen-

t i d y closed under gaph composition, a result which we then exploit to show that

the real roots of independence polynomials have cloçure (-w, O], while the complex

roots are dense in @, even for some restricted families of graphs. We then consider, for

an arbitrary graph G, what happens to the independence roots of the graphs G[G],

(G[G])[G], and so on. The independence polynomials are iterates (with respect to

hinction composition) of i(G, x) (the independence polynomial of G), and their mots

approach (in a very strong sense) the Julia set of i(G, x) - 1, thereby associating with

C a fractal. 1(G), which we cal1 the independence attractor of G. We are led to ask:

when is 1(G) connected? For graphs with independence number 2, as well as a few

infinite families of graphs with arbitrarily high independence numbers, we provide a

complete answer to the question.

Chapter 2

Roots of Chromatic Polynomials

The chromatic polynomial, n(G,x), of a graph G is the polynomial whose value at

each positive integer z is the number of functions f : V 4 {l.. . . J } such that

UV E E implies f (u) # f (v), where V and E are the sets of vertices and edges of G'.

respectively. The roots of sr(G,x) are often called the chromatic roots of G. and in

general a chromatic root is any (cornplex) number which is a root of some chromatic

po'ynomial. Initial interest in chromatic roots stemmed from the former Four Colour

Conjecture. which states that no planar graph has 4 as a chromatic root. The study

of chromatic roots has since emerged as a rather fascinating topic in its own right,

having attracted considerable attention (cl. [9, 11, 18, 20, 21, 22, 45, 50, 57, 601).

However, much remains unknown.

We s h d need a few basic facts about chromatic polynomials (d of which are

discussed in [IO]), which we review now.

Let G be a graph, possibly containhg parallel edges or loops. Pardel edges have

no effect on the chromatic polynornial, and it follows directly from the definition that

if G has a loop then indeed *(Gy x ) = O. If G has no loops, *(G,x) is a monic

polynornial in x of degree IVI, whose coefficients are integers that altemate in sign.

The well known deletion-contraction reduction states that for e any edge of G, n(G,x) = *(G - e , x ) - r (G a e , x ) , where G e is the contraction of e in G, and

is obtained by removing e and identifying its end vertices. It is not hard to verify

that the formula holds even if e is a paraliel edge or loop. We sometimes rewrite the

deletion-contraction T(G - e, x) = a(G, z) + T(G e, x), which we will refer to as

addition-contraction.

Also, if two graphs G and H intersect exactly on a complete graph, Kp, on p

vertices, then x(G u H, x) = *(G, x)*(H, x ) / ~ ( h ' , , x); t his is sometimes referred to

as the Complete Cube t Theorem.

Finally, the chromatic polynomials of Kn, T, (any tree of order n), and C, (the

cycle of order n) are given by x(x - 1) O - - (x - n + 1), x(x - 1)"-', and (x - 1)" + ( - l ) * ( x - 1), respectively.

Here is an example of a chromatic polynomial calculation; let G be the graph

shown below:

We can use deletion-contraction and the complete cutset theorem to calculate

r (G, x ) as follows:

One avenue of investigation into chromatic roots has been determining bounds on

their moduli in terms of various graph parameters. Very recently, Sokal proved the

following, which c o h a 1972 conjecture of Biggs, Damereu, and Sands [II]:

Theorem 2.0.1 ([54]) If G is a graph of maximum degree k , then euery chromatic

mot z of G lies in the disc lzl < 7.963907 k.

While the constant 7.963907 found in [54] can likely be improved, the linearity of the

bound is best possible, since the complete g a p h Kk+l (which has maximum degree

k) has a chromatic root at z = k.

A complementary approach is to bound the chromatic roots in t e m of the co-

mnk (or cycle rank) of the graph. Recall that a connected g a p h with n vertices and

m edges has corank m - n + 1. Brown has has recently proved:

Theorem 2.0.2 ([21]) If G is a graph of wmnk k 2 1, then every chromatic mot t

of G lies in the disc Ir - 11 5 k.

Unlike the bound in te- of maximum degee, however, it is not known whether

linear gowth with corank is best possible.

It is natural to ask for suitably restricted subcIasses of graphs which satîsfy a

sublinear bound in either maximum degee or c*rank, and indeed in [29] a sublinear

bound on the chromatic roots of 60th corank and maximum degree was proven for

graphs called generdized theta graphs. Our numerical explorations with theta graphs

suggest an interesthg conjecture which extends that result, and in section 2.1.1 we

prove our conjecture for 'small' theta graphs.

Roots of chromatic polynomials lying in the leRhalf plane have also attracted

interest, and very little is known about them. In section 2.1.2 we prove that indeed

idbitely many chromatic roots with negative reai part are achievable among theta

gaphs of CO-rank 2, thereby providing a family of minimal corank having chromatic

roots with negative real part.

While theta graphs may be a very restricted subclass of gaphs*, we prove in

section 2.2.4 that having a theta subgraph is both necessary and sufficient for a

graph to possess a subdivision whose chromatic polynornial has a root with negative

real part. In section 2.2.3 we are able to describe the limit points of the chromatic

roots of 'uniform' subdivisions of a graph. and finally, in section 2.2.5 we prove that

large subdivisions of a.ny g a p h draw the chromatic roots close to the disk 1 z - 11 5 1.

2.1 Generalized Theta Graphs

For a positive integer k, the k-ary generalired theta graph El,,,.--,,,, is formed by taking

a pair of vert ices u, u (called the endvertices) and joining them by k internally disjoint

paths of lengths si,. . . , sk 2 1. The adjective, 'generalized', is more philological than

mathematical, and we will often &op it in what foilows. For brevity, we denote by

8(t;s) the k-ary theta graph whose path lengths are dl equal to S.

Let us derive an expression for the chromatic polynornials of generaüzed theta

gaphs. Observe that if one adjoins to a generdied theta graph 8,,,...,s, a new edge

e between the two endvertices, the resulting g a p h is a collection of cycles (of lengths

si + 1) that overlap in a complete graph K2 (namely the edge e), so that

Likewise, contraction of the endvertices of O,,,..-,s, yields a collection of cycles (of

'Sokal has tecently proved that chromatic mots of theta graphs are dense in al1 of C, except possibly the disk 1% - II 5 1. Thus, theta graphs are not such a "restricted" family after ait.

lengths s i ) that overlap in a complete g a p h of order 1, so that

It follows that the chromatic polynomial of 8,,,...,s, is given by

k

[(x - + ( - ~ ) ~ * + l (x - l)]

- (1 - s)"l n [(1 - - 11 1 .

Therefore. we need only concern ourselves with the roots of

where y = 1 - x. AU of our calcdations in section 2.1.1 will be expresed in terms of

the vasiable y.

Let us now dispose of some trivial cases. If k = 1, the theta gaph 8,, is isomorphic

to the path P,,, so that its chromatic roots are O and 1. If k = 2, the theta gaph

8.,, is isomorphic to the cycle Cs,+,, , so that its chromatic roots (other than z = 1)

ail lie on the circle lz - 11 = 1. We s h d therefore assume henceforth t hat k 2 3.

If one or more of the path lengths si equals 1, then the second product in (2.4)

vanishes, and ai l the chromatic roots (other than r = 1) again Lie on the circle Iyl = 1,

Le. 12 - 11 = 1. We shall therefore assume henceforth that a l l path lengths si are at

The above metbod for

and Tutte [51, pp. 29-30]

computing ~(8,,,...,~,, z ) was employed previously by Read

for the case k = 3.

2.1.1 Theta Graphs with 5 8 Paths

A k-ary theta graph clearly has maximum degree k and corank k - 1 (except for the

trivial case k = 1 with si 2 2, which has maximum degree 2). The main result of [29],

stated below, thus proves that the chromatic roots of the family of all k-ary theta

gaphs are bounded sublinearly in both cwank and maximum degree.

Theorem 2.1.1 ([29]) The chmmatic roots of any k-ary generahed theta g ~ u p h lie

in the disc lz - 11 5 [l + ~ ( l ) ] k/ log k . where o(1) denotes a constant C ( k ) that tends

to zero as k 3 00.

This bound is asymptotically saturated by the graph 0(L;2) with all path lengths equal

to 2 (which is isomorphic to the complete bipartite graph K2,k).

Now, numericd cornputations suggest that among a l l k-ary theta gaphs Q,,,...,3,,

the one wit h a chromatic root that rnaximizes lz - 11 is the graph with al1 path lengths

si equal to 2, Le. the gaph 0(k:2) cz fiVh. We conjecture thst this is indeed the case.

The general bound of Theorem 2.1.1 is not strong enough to prove this conjecture.

Nevertheless, by different techniques we shall show the validity of this conjecture for

all k 5 8. As before, it s d c e s to consider k 2 3 and si, . . . , sk 1. 2. Denote by

p(s1, . . . , sk) denotes the maximum modulus of a root of f, ,,,.., 3, (y).

Our method is based on the following trivial bound (see e.g. [47, Theorem 27.11):

Proposition 2.1.2 Let P(y) = 2 ajv be a polynomiai of degree n (so that a,, # O ) , j=O

and let R be the unique nonneqative real solution of

Then ail the roots of P lie in the disc lyl 5 R. Moreover, in (2.5) the numûers lajl

for O 5 j 5 fi - 1 can be replaced by any numbers bj 2 laj 1; this only makes the bovnd

At first sight it is surprising that such a crude estimation method - which throws

away aii the sign or phase information in the coefficients of P - could yield reasonably

sharp results. And indeed, we do not entirely understand why it works so well in our

application - but it does! Here is the key trick: since the polynomial fs,,-. . ,sk(y)

defined in (2.4) is divisible by (y - I ) ~ , we are free to puIl out a factor (y - 1)' with

anyO<l lk to take 1 = 1.

before applying Proposition 2.1.2. It tums out that the right choice is

That is, we define the polynomial 4 ,,,...,,,(y) by

f ~ l ~ - + - ~ ~ k (Y 4 s ,,.... 3, (Y = y - 1 (2.6)

Let [k] denote the set {l, . . . , k}, and let (1) denote the set of al1 subsets of [k] of

cardinality 1. For a subset X = (il, i2. . . . . il} of [k]. let us define

We then have:

k k

= C(-1)" 1 y'" - C(-qm 1 m=O X Ç ( ~ W ) m=O XC(~!?)

and hence

We cm now implement Proposition 2.1.2 by defining h,, (y) to be the poly-

nomial obtained from t$s,,...,3,(y) by changing aii subleading signs to - as in (2.5),

and letting r(sl ,..., sk) be the unique positive root of h, ,,..-, s,(y). We then have

p(si,. . . sb) 5 r(sl ? . . . sk)) where (recd) p(sl,. . . , sk) is the maximum modulus of

a root of f,,,...,s, (y). Unfortunately, this bound is unsuitable for our present purposes,

as it turns out that r (s i , . . . , s k ) is not a monotonicdy decreasing function of the

pat h lengths SI, . . . , s k . We t herefore t hrow away a bit more, by disregardhg all sign

cancellations among the subleading terms of (2. I l ) , and define

[Thus, the coefficient of yJ in (2.12) is in general Iarger in magnitude than in (2.11).] - Let T(sI . . . . , s k ) be the unique positive root of hs,,...,s, (y). Then it follows immediately

from Proposition 2.1.2 that

or in other words:

Proposition 2.1.3 Euery chromatic root z of Os,l...,s, lies in the dix lz - 11 5 F ( s ~ , - . . ,sk).

We now analyze the behaviour of the upper bound F(sl , . . . sk):

Proposition 2.1.4 7(si,. . . , sk) is symmetric in si,. . . , sk and stn'ctly demdng in

each si.

Proof The symmetry is obvious. To prove the decreasing prope*, fix si, . . . , q and - set r = qsll s2). . . , sk); by symmetry, it sufIices to show that F(sl + 1, sl, . . . ,si) < r. Now, equation (2.12) implies that r > 1. F'rom equation (2.12) we can rewrite - h, ,,,,... &(Y) in the form

where A and B are poiynomials in y with nonnegative integer coefficients that do not

depend on s i . Moreover, the degrees of y s ~ A(y) and B(y) are less than (CL1 s i ) - 1 ,

and B(0) = 1 . It follows that r(x:=i - rSIA(r) > O . Now from (2.14) we have

It follows that 7(sl + 1, s ? ~ . . . , s k ) < T = ?(si! s2,. . . s i ) , completing the proof. O

What rnay be surprising is how well the mots of h and h bound the roots of f for

small k (see Table 2.1). In particular, they are good enough to prove the following.

Theorem 2.1.5 For 3 5 k <_ 8 , we have p( s l , . . . , si) <_ P(k ;2 ) , Wth equality on$

when sl = - . . = sk = 2. In other words, among al1 k-ary theta graphs, the graph vieth

a chromatic root that maximzzes )z - 11 is the one zuith all path lengths equal t o 2.

Proof By direct calculation (see Table 2.1) we have i ( 2 , 2 , 3 ) < p(2 ,2 ,2 ) , F(2,2 ,2 ,3) < - p(2 ,2 ,2 ,2) and 3 2 , 2 , 2 , 2 , 3 ) < p ( 2 , 2 , 2 , 2 , 2 ) . The result for 3 k 5 5 then fo110ws

immediately from Propositions 2.1.3 and 2.1.4. For k = 6,7, a bit more work is

needed: the calculations show that p(2, . . . ,2,2,3), F(2, . . . ,2,2,4) and T(2, . . . ,2,3,3)

are a l l bounded above by p(2, . . . , 2 , 2 , 2 ) , so that the result again follows from Prope

skions 2.1.3 and 2.1.4. Finally, for k = 8, the calculations show that p(2, . . . ,2,2,3),

p(2, . . . ,2,2,4), F(2, . . . ,2,2,5) and F(2, . . . , 2 , 3 , 3 ) are all bounded above by

p(2, . . - ,2,2,2), which is again sdicient. O

This method of proof relies in an essential way on the fact that, for 3 5 k

8, only finitely many of the upper bounds F(si,. . . , s k ) are larger than the true

Path length sequence Actual value &)(SI, . . 1 sk)

1.5247025799 1.3247179572 1.9635530390 1.6 180339887 2.3602010481 1.9596554O46 1.9125157044 2.0227195761 1 Mg2237868 2.7305222731 2.3291754791 2.3208606055 2.0524815723 3.0823336669 2.6933092033 2.7030241913 2.3573224846 3.4201564280 3.0446178232 3.0625912820 3.0953618332 2.6885399588 3.746884928 1 3.3836067543 3.4054981704 3.4292505541 3.4182415134 3.4200422197 3.4203605983 3 A2OO94773l 3.4201605551 3.4201602535 3.4201547358 3.4201566935 3.0254986086

Upper bound

Table 2.1: Values of p(sl, . . . , sk) and its upper bounds r ( s l , . . . , s k ) 5 7(si,. . . , sk).

value p(2 , . . . ,2) = P(k;2). Unfortunately, this fails for k = 9: indeed, we have

iim F(sl, . . . , S~-I, sk) = F(sl,. . . , sk-1) a d in pasticdar Lm F(2,. . . , 2 , s9) = F@;$) = S k d o a ug-00

3.7959050193 > 3.7468849281 = p(g;z:2). A genuinely new method, therefore, seems to

be required to prove Theorem 2.1.5 for k 2 9.

2.1.2 Theta Graphs with 3 Paths: An Explicit Family having

Roots with Negative Real Part

Since the coefficients of any chromatic polynomial altemate in sign, no real root of a

chromatic pol-vnomial is negative. But can a chromatic root have negative real part?

Based on the chromatic roots of al1 graphs on at most 8 vertices, the foilowing was

conjectured in 1980.

Conjecture 2.1.6 [33] There are no chromatic rwts wi2h negatzve real part.

In 1991 Read and Royle computed the chromatic polynomials of ali 3-regular

graphs on at most 16 vertices, noting that graphs with high girth appear to be

contributing the roots with smdest real part. They proceeded to plot the chromatic

roots of al1 3-regular graphs of girth at least 5 on 18 vertices, and observed the

following, thus providing the smallest known counterexample to the above conjecture.

Proposition 2.1.7 [50] There ore graphs of order 18 having a chromatic root &th

negatzve real part.

They noted the same for the 3-regular gaphs of girth at least 6 on 20 vertices,

and girth at les t 7 on 26 vertices. More recently, Shrock and Tsai [53] have shown

how badly the conjecture fails by showing that as k -, oo, the k-ary theta graphs (i.e.

graphs fomed from two vertices joined by k intemally disjoint paths) had chromatic

roots whose real parts tended to -00. By different techniques (namely the Hermite-

Biehler and Sturm theorems on the roots of real polynomials), we show here that

indeed infinitely many chromatic roots with negative real part are achievable among

3-ary theta gaphs themselves. These provide examples of the srndest corank.

To this end, we restrict our attention now to the subfamily {Oqa,, : a > 2) of generalized theta graphs whose u-u paths dl have the same length. The smallest such

graph having a chromatic root with negative real part is found (by direct calculation)

to be B8,e,8. We c m Say much more.

Theorem 2.1.8 For a 2 8 the graph 8,,, hm a chromatic root with negative real

part.

Proof Setting k = 3 and sl = s? = s3 = a in equation (2.3)? we find (after a Little

simplification) t hat

For a 2 8, we need to show that ~ ( e ~ , ~ , ~ , -x) has a root with positive real part. Le..

t hat

&(I) (1 + x)~'-' - 3(1+ x ) ~ + 2 + 3: (2.17)

is not Hurwitz quasi-stable. So let a 2 8, and expand +,(x) into its even and odd

parts:

Now set t = x2 (as in Theorem 1.2.2). Several calculations suggested that Pt( t ) always

appears to have a nomeal root (for a 2 8), and by Corollary 1.2.3 of the Hermite-

Biehler Theorem (Theorem 1.2.2), it is enough to show that this is indeed the case.

To that end, it would suftice (by Theorem 1.2.4) to show that its Sturm sequence

contains either a negative leading coefficient or gaps in degree. Our computations,

however, suggest that this does not occur until close to the end of its Sturm sequence.

So let us instead consider the polynoniial

which clearly has a nonred root if and only if e(t) does. Moreover, we can establish

the following.

Lemma 2.1.9 For a 1 14, the Sturm sequence of &(t) has as its fiph t e m o poly-

nomzal uith negative leading coeficzent.

We shall see, also, that before the fifth term in the sequence, there are neither

gaps in degree nor any negative leading coefficients. However, since Lernma 2.1.9

tells us the leading coefficient of the fifth term is negative, we conclude that da( t ) has

a nonreal root for a 2 14, and in fact for a 2 8 upon venfying the cases 8 5 a 5 13

directly. Hence, to prove Theorem 2.1.8, it remains only to prove Lemma 2.1.9. We

assume that a 2 14 is even (the odd case is handled similarly). Then kom (2.17),

(2.18), and (2.19) we find that

Let us denote the fvst five terms of the Sturm sequence of @a by &O (= #J,

4 ( = 4 ) # , d # Then it is clear, from the form of 4. and the division

process, that lcoeff (4:) is a real valued function of a, which we shall show is always

negative (for a 2 14 even). We s h d see that only the k s t 7 terms of @a are needed.

To begin, we have

3a 1 3a-1 3a- 1 whereb= ( ;) - 3 ( i ) , c = ( , ) - 3 ( f ) , ..., h = ( ,,) -3(P,), a n d n = " 2 .

Note that b, c, . . . , h and n are polynomials in a with rational coefficients. Now

Dividing & by $:, we find

Since bn2 > O, we can (by an observation made in Section 1.2.2) clear the denomina-

tors by choosing 4: = bn2 - (-rem(#a, 4;)). Now it turns out that c2(n - 1) - 2bdn is

always positive (for a 2 14 even) . Indeed, it is a polynomial in 6, c, d, and n wit h in-

teger coefficients, each of which (recd) are themselves polynomials in a with rational

coefficients. Ca-g out the substitutions in Mapie, we obtain an exact expression

for c2(n - 1) - 2bdn E Q[a], and find that it has positive leading coefficient, and so

is positive beyond its largest real root, a bound for which is obtained by applying a

standard result (cf. (30, p. 1971) to the polynomial. It is then verified directly that

the polynomid c?(n - 1) - 2bdn is also positive for those (even) values of a between

14 and that bound.

Moving on to the next term in the Sturm sequence. we divide & by &, and find

where

u = - (c2(n - 1) - 2bdn) {d(n - 2) (c2(n - 1) - 2bdn)

-h (ce(n - 3) - 46 f n)}

+ (cd(n - 2) - 3ben) {c (n - 1) (c2(n - 1) - 2Mn)

-h (cd(n - 2) - 36en)) ,

and

-h (cg(n - 5) - 6bhn))

+ (cf (n - 4) - 5bgn) {c(n - 1) (c2(n - 1) - 2bdn)

-bn (cd(n - 2) - 3ben)) .

Multiplying by the positive number (c2(n - 1 ) - 2bdn)*, we choose

Again, we have verified (with the aid of Maple) that indeed u is always positive (for

a 2 14 even). Let us move on to the 6fth tenn in the Sturm sequence. Dividing 4: by &, we find

-u {(ce@ - 3) - 4b f n ) u - (c2(n - 1 ) - 2bdn) w ) ) tn-''

Multiplying by the positive number u2, we choose

We denote by lcoeff (4:) the coefficient of tn-4 in #: (no conhision arises in doing so,

as we are about to see that this coefficient is never zero, and so really i s the leading

coefficient of 4:). So lcoeff (4:) is a polynomial in b, c, . . . , h, and n with integer

coefficients. Substituthg (once again) our expressions for b, c, . . . , h as polynomials

in a with rational coefficients, we obtain an exact expression for lcoeE (4:) E Q[a],

the fmt few terms of which are approximately

In particuiar, the fmt term is negative, and so lcoeff (4:) is negative for a sdliciently

large, which is what we want. Applying a standard result (c. t [30, p. 1971) to the

polynomial lcoeff(4:), we obtain 134 as a bound on its largest real root, and so the

proof of Lemma 2.1.9 for a even is completed by v e r b g directly that lcoeff (@f) is

also negative for a = 14,16,18,. . . ,134. We mentioned that the proof for a odd is sirnilar. There we find

lcoeff (4:) = -342.7311661am + 21277.83225as9 - 617481 .3554a6& + . - ,

in which case an analysis like the previous paragraph is canied cut.

This completes the proof of Lemma 2.1.9, and so of Theorern 2.1.8. 0

We remark that the smdest generalized theta graphs (in terrns of number of

vert ices) having a chromat ic root wit h negat ive real part are 84,s,5,5,5 and e5,6,6,6i each of order 2 1.

2.2 Large Subdivisions of Graphs

We constructed t heta graphs by taking a &- bond and replacing its edges by pat hs.

In general, if e is an edge of a graph G, then by subdividing e we mean replacing

e by a path. A subdivision of G is any graph formed by subdividing (one or more)

edges in G. It is natural to ask what effect this operations has on the roots of the

chromatic polynomial. In Section 2.2.2 we shall derive an expression for the chromatic

polynomials of subdivisions of G. This expression simplifies considerably in the case

of unzform subdivisions of G (cl. Section 2.2.3). Their chromatic polynomials form

what is known as a recursive family of polynomials, and we will apply a theorem of

Beraha, Kahane, and Weiss (81 to describe the limits of their roots. We will see that

the circle lz - 11 = 1 plays a key role in describing the b i t s of chromatic roots of

d o m subdivisions of a graph, and is itself among those limits (Figure 2.1 shows

the roots of a subdivision (a theta graph with paths of length 10, 10 and 11) of K4 - e

dong with the circIe Ir - il = 1).

We wiU derive two interesting consequences of our work here. Firstly, recd that

in the previous section we determined the family of graphs with minimal cerank

Figure 2.1: The chromatic roots of a subdivision of K4 - e wit h I Z - 11 = 1.

having chromatic roots with negative real part (some fùrther graphs having chromatic

roots with negative real part are provided in [50, 52, 53, 551). A corollary to our

expansion of chromatic polynomials of uniform subdivisions of graphs leads to a

complete characterization of those graphs which have a subdivision having a chromatic

root with negative real part.

Secondly, experimental evidence of chromatic mots of subdivisions leads to the

observation that the roots tend to be drawn towards the unit circle centered at t = 1.

We show that in fact for any E > 0, the chromatic roots of al2 large subdivisions of a

graph have their roots in Ir - 11 < 1 + E (irnproving a recent result [20] which only

proves some subdivision has its roots in this region).

2.2.1 Recursive Families of Polynomials

Before we proceed onto a discussion of the mots of chromatic polynomials of subdi-

visions of graphs, we need to state (in detail) an analytic result on particular families

of polynomials (namely, recursive familes). We begin with the foilowing definition.

Definition 2.2.1 If {f,(x)) zs a famzly of (cornplex) ppolynomials, tue say that a

nurnber r E C 13 a lzmit of mots of { f,(x)) if either f&) = O for al1 suficiently large

n or z U a lirnit point of the set R({fn(+))), where R({f,,(x))) is the union of the

mots of the f,(x).

Now (as in [SI) a family { fn(x)) of polynomials is a recursive family of polynomials

if the f&) satisb a homogenous 1inea.r recurrence

where the ai(x) are fixed polynomials, with ak(x) $ O. The number k is the order of

the recurrence.

We can form from (2.20) its associated characteristic eqvation

whose roots X = X(x) are algebraic huictions, and there are exactly k of them counting

mdtiplicity (c.f. [l, 431).

If these roots, Say Xl(s), A2(x), . . . , Ab(x), axe distinct, then the general solution

to (2.20) is known [8] to be

with the 'mal' variant (cf. [8]) if some of the Ai(x) were repeated. The huictions

al(x), q(x ) , . . . , a k ( x ) are determined from the initial conditions, that is, the k linear

equations in the ai(x) obtained by letting n = 0,1,. . . k - 1 in (2.22) or its variant.

The details are found in [BI.

Beraha et al. [8j proved the resuit below on recursive families of polynomials and

their roots.

Theorem 2.2.2 ( (8) ) If { f n ( x ) ) is a recursive family of polynomials, then a complex

number r is a Izmit of roots of {fn(z)) if und only if there is a sequence {G} in 6

svch that f,(z,,) = O for al1 n and Zn -+ z as n + CQ.

The main result of their paper characterizes precisely the limits of roots of a

recursive h i l y of polynomials.

Theorem 2.2.3 ([BI) Under the non-degeneracy repirements that in (2.22) no a i ( x )

is identicallgl zero and that for no pair i # j is & ( x ) d , ( x ) for some complex

number w of unit modulus. then z E 63 is a lzmit of roots of { f&)} if and only zf

either

(i) two or more of the Xi(2) are of equal modulus, and strictly greater (in modulus)

than the others; or

( i i ) for some j . & ( z ) has modvlus strictly greater than all the other A&) have, and

aj(2) = 0.

This result has found application to the chromatic roots of recursive families of

graphs ( c f . (1 Il), that is, families of graphs whose Tutte (and therefore ckomatic)

polynomiah satisfy a homogeneous ünear recurrence; see (9, 501 for some examples.

It is also proved in [8] that the kst non-degeneracy requirement in the statement of

the theorem is equivalent to f,(x) satiswg no lower order (homogeneous, lineu)

recurrence. What we s h d need here is the following.

Coroiiary 2.2.4 Suppose {f,(x)) Ls a famdy of polynomiols svch that

where the ai(x) and X i ( $ ) are j2re.d nonzero polynomâals, such that for no pair i # j d &(x) wXj(x) for some u E @ of unit modulus. Then the limit~ of T O O ~ S of { f,(~)) are exactly those z satisfying ( i ) or (ii) of Theorem 2.2.3.

Proof It is enough to show that f,(x) satisfies a k-th order homogeneous linear - recurrence, for then Theorem 2.2.3 applies as the non-degeneracy requirements are

satisfied here. Such a recwence is:

together with the initial polynomials

where the a i (x ) are such that

This completes the proof.

In the next section, we derive an expression for the chromatic polynomials of

subdivisions of a graph, that when restricted to the uniform case, provides a recursive

family of polynomials. The Beraha-Kahane-Weiss Theorem wiil then allow us to

derive some precise information on the limit points of these chromatic roots.

2.2.2 An Expression for the Chromatic Polynomials of Sub-

divisions

Let u s assume, for the remainder of the chapter, that G is a graph, urithout loops, which

may indeed have parallel edges; its vertex and edge sets V and E have cardinalities

n and rn, respectively. Also, any parallel edges andior loops resulting £rom the

contraction of an edge at any time are @ to be thrown away.

We derive now a rather technical expression of the chromatic polynomial of a

generd subdivision of a gaph. What is crucial is the expansion of this polynomial in

terms of powers of 1 - x and coefficients that depend only on the underlying graph

(and not the exact subdivision we have taken).

Theorem 2.2.5 Let G = {el, . . . e r ) C E, and G;:"*'C B.--* the graph obtazned fron G by svbdividing edge e, into a path of length Zi (z = 1,. . . , k). Then

- C fil, ..., ik -1 ( x ) ( I - x;:; lt,

where

and the f 's (and clearly gEt) are polynomials Mat depend on G and E' = {el,. . . , e k ) ,

but pJ on Il, ..., it.

This can be proved by induction on k. Let us e x d e the cases k = 1 and k = 2,

the latter being sdciently descriptive of the general argument which is tedious but

no more difEcult. Because of the degree of symbolism involved, it will be convenient,

for the remainder of t his section only, to denote the chromatic polynomial n ( H , x) of

a graph H by the symbol H itself, and it will be clear from the context whether we

are actudy referring to the graph or its chromatic polynomid.

For k = 1, we are subdividing a single edge, e, of G into a path of length 1, Say.

Tossing e into the graph Gi, and contracting, we have

Now G; + e creates a cycle of length 1 + 1, intersecting G exactly on e, while

(Gf + e) a e produces a cycle of length 1 which intersects G e on a single vertex.

Hence

which establishes the result for k = 1.

Now for k = 2, we want an expression for the chromatic polynornial of G$ the

graph resiilting from subdividing edges e and f of G into patbs of length 1 and s,

respectively. This we derive €rom the case k = 1 (more specificdy, from (2.25)):

ce*f = (ci)! = - 1,s

(-')' {(ci + GE f)(i - z)" - (Gr + (1 - x)G~ f)) x

and

Hence, we have

-(G + G . e + (1 - x ) ( G f +Ga e))(l- x)'

The 'coefficient' of (1 - x)"+' above is exactly G - e - f, as G + G e = G - e

and G f + G f e = (G f ) - e, and clearly (G f ) - e = (G - e) f, giving

This estabüshes the case k = 2 £rom the previous case (k = 1). 0

We will see that Theorem 2.2.5 has some deep consequences in terrns of the location

of chromatic roots, especially when restricted to the natural subfamily of uniform

subdivisions.

2.2.3 Uniform Subdivisions and the Lirnits of their Chro-

matic Roots

When subdividing each edge of E' the same number of times, Theorem 2.2.5 specid-

izes to the foilowing.

Theorem 2.2.6 Let E' = {el,. . . ek) C E, and GY the p p h obtazned from G by

subdividing each edge of Et znto a path of fength 1. Then

where g ~ t iS given by (Z.24), and the f 's (agazn) are polynomials that depend on G

(and G) but & on 1.

The key point is that expansion (2.26) expresses n(Gf', x) as a recursive family.

and hence we can employ the power of the Beraha-Kahane-Weiss Theorem. In doing

so, we find al1 of the iimit points of the uniform subdivisions {G? : 1 2 1) of G

(without explicitly hd ing the chromatic roots of each of these graphs!). CY

We need yet one bit of technical notation; G will denote the edges of E' t hat are

not bridges.

Theorem 2.2.7 I,f Et is a subset of E containing at lead one edge that is not a

bridge of G , then the limits of th.e chromatic mots of the family {Gy) are exactly

( i ) the c i~c l e Ir - 11 = 1:

hr

(ii) the mots of r(G- Et, I) outside Ir - 11 = 1, and

(iii) the roots of gs (x) inside Ir - 11 = 1.

Proof Let us fint examine the case where G contains no bridges, in which case - N

Et= Er. Clearly *(G - G , x) is not identically zero. And neit her is g ~ t (x) , for suppose

gEt (x ) E O. Then, kom (2.26), we would have t hat (1 - x ) ' divides G: . However , i t

is well known (cl . [60]) that the multiplicity of 1 as a chromatic root of a graph is

the number of blocks in the gaph. Since Er contains no bridges, for each 1 the gaph

GF' has the same number of blocks as G, so that the multiplicity of 1 as a chromatic

root of Gf' cannot possibly go to infinity with 1, a contradiction.

Now ignoring the factor JS- in (2.26) and rewriting (-l)kgEl(z) as

(- l)kgEt (x) l', we can apply Corollary 2.2.4, and therefore (i) and (ii) of Theorem

2.2.3, to get the limits of the chromatic roots of {CF).

Applying (i) of Theorem 2.2.3, we irnmediately get the circle Ir - 11 = 1 as b i t s ,

by setting all of the IA,(z)l equal, i.e.,

This is the only situation where (i) of Theorem 2.2.3 gives any Limits, for setting

any fewer than a l l of the IAi(z)I equal here will, upon applying (i), amount to finding

values r such that lz - 11 = 1 and lz - 11 > 1, which is impossible.

Moving on to (ii) of Theorem 2.2.3, one application gives the roots z of n(G- E', x)

such t hat

that is, the roots z of n(G - E', x) such that 11 - rl > 1.

Another application of (ii) gives the roots z of g ~ t such that

that is, the roots z of g ~ t such that 11 - zl < 1. Finally, applying (ii) to any j such that O < j < k would amount to finding roots

z of fj such that 11 - zl < 1 and (1 - zl > 1, which is clearly impossible.

Now in the case where E' does contain some bridges, subdividing a bridge e E E'

is merely the replacement of a bridge by a path, from which it follows (quite easily.

using the complete cutset theorem) that *(GY, x) = (x - l ) ' - ' s ( ~ f - ~ , z). Repeating

this argument recusively over all the bridges in E', we have that the limits of the Cy

chromatic roots of {~f') are exactly those of {G:'}, £rom which the result foilows

kom the previous case. This completes the proot 0

N

When G = E, it is clear that G- E' is a forest, therefore having no chromatic

roots outside It - 11 = 1. Also, we write Q (z) instead of g E ( z ) .

Corollary 2.2.8 Let Ci be the graph obtazned from G by subdividzng each edge into

o path of length aactly 1. Then, if G is not a forest, the limits of the chromatic rwts

of the famdy Gr are ezoctly:

(i) the czrcle lz - 11 = 1;

(ii) the mots of 9 (z) inside Ir - 11 = 1.

Based

las, 2.2.8

on direct calculation for various small graphs, it appears that (ii) of Corol-

sirnply never happens, except of course for the point z = 1 which is clearly *Y

a root of 2 (2). In fact, we conjecture t hat if t is a mot of g (z), then Ir - 11 is either

O or 1.

2.2.4 Application 1: Chromat ic Roots wit h Negative Real

Part

Our first application of our investigation of subdivisions concerns chromatic roots

with negative reai part.

Very little is known about the chromatic roots lying in the left-half plane. Earlier.

we pointed out that it was conjecttued [33] in 1980 that in fact there are none at all,

and Rcad and Royle [50] showed recently by direct calculation with cubic graphs that

t hey do exist . In section 2.1.2. we proved the existence of infinitely many chromatic

roots with negative real part, and that there are graphs of arbitrarily high girth with

them. More precisely, we showed the graph El,,=, has a chromatic root with negative

real part for each a 2 8, and that the moduli of these roots get arbitrarily small. (The

existence of infinitely many chromatic roots with negative real part was demonstrated

independently in [52, 551 by ent ireiy different met hods. )

Of course, generalized theta graphs are very specific graphs. But it turns out they

are the key ingredient in characterizing the gaphs in general which have a subdivision

having a chromatic root with negative real part. It is hom Theorem 2.2.7 that we

are able to make this connection, and it is perhaps a bit surprising that most graphs

have a subdivision having a chromatic root with negative real part. We say that G

has a Bsubgraph if G has a subgaph isomorphic to some generalized theta graph.

Theorem 2.2.9 The followkzg are quivalent.

(i) G hm a subdivision having a chromatic mot with negatzue real part;

(ii) G has a block of CO-rank at least 2;

(iii) G hm a 8-subgmph.

Proof We prove the implications (i) (ii) =+ (iii) + (i). -

(i) (ii). It is easy to see that blocks of CO-rank O are trees, and of CO-rank 1 are

cycles. Thus, if no block of G has CO-rank larger than 1, then the blocks of G are

simply bridges and cycles. Hence the blocks of any subdivision of G are K2's and

cycles as well. And since neither K2 nor any cycle has a chromatic root with negative

real part, neit her do the subdivisions of G.

(ii) (iii). Without loss of generality, assume that G is 2-comected and has C O - r d

at l e s t 2. Let e be an edge of G. Clearly e is not a bridge, and therefore lies on a

cycle, Say Cl, of G. Now G rnust have an edge, say el, other than those of CI (or

else G = Cl and G has CO-rank 1). Since G is 2-connected, e and el both belong to

a cycle C2(# Cl) and e E Ci n C2. Now, Ci together with a component of CI @ C2 (the symmetric daerence) is a B-subgraph of G? as required.

(iii) * (i). From the complete cutset theorem. it is enough to show the result holds

for 2-connected graphs. So we assume G is 2-connected. Start with a 8-subgraph

of G, and subdivide it into e, , . for some h e d a 2 8, obtaining a subdivision H of

G containhg as a subgraph Ho = eaVqa. Now let E' be the edges of H that do not

Lie on Ho. Then H - E' consists exactly of Ho and possibly some isolated vertices, a

graph which we know (cf. section 2.1.2) has a chromatic root with negative real part.

Thus if Er is empty, then we are done. And if not, then consider the family {HF'). Since E' has no bridges and H - E' has a chromatic root z with negative real part,

then by (ii) of Theorem 2.2.7, z will be a k t of the chromatic roots of (HF'). The

HF'% are indeed subdivisions of G, and it follows that for I sufbciently large, HF has

a chromatic root with negative real part. 13

From Theorem , we irnrnediately deduce the following.

Corollary 2.2.10 Every non-empty graph hm a series-pa~allel extension having a

chromatic mot with negatzve real part.

We can sometimes make use of subdivisions to generate, from a single chrornatic

root with negative real part, an infinite cluster of such chromatic roots. For suppose

e is an edge of G which is not a bridge, and that G - e has a chromatic root z with

negative real part. Then, from (ii) of Theorem 2.2.7, z is a limit of the chromatic

roots of the family {Gr}. If, in addition, r is not a chrornatic root of Ga e, then, from

(2.25), r will not be a chromatic root of any Gr, and so in fact must be a h i t point

of the chromatic roots of the family {Gf). Together with Theorem 2.2.2, we find a

sequence { z l } in C \ { z ) such that r(Gf, z l ) = O and y 4 z.

Fix any a 2 8, for instance, and let F = 8a,a,a+uv, where u and v are the terminais

of Then F - UV = 8,,,,,, which we know ( from Section 2.1.2) has a chromatic

root z wit h negative real part, while F UV is jus t three cycles intersecting on a single

vertex, a graph having no chromatic roots with negative real part whatsoever, as the

chromatic roots of cycles lie on the disk lz - 11 = 1. Thus {4uv) is a family of gaphs

producing infinitely many chrornatic roots with negative real part.

2.2.5 Application 2: Bounding the Chromatic Roots of Large

Subdivisions

Our second application of o u . investigation of chromatic roots of subdivisions centers

on the location of the roots of large subdivisions of a graph. A few plots (see, e.g.,

Figure 2.1) of the chromatic roots of subdivisions of several s m d graphs will indicate

that subdividing edges tends to draw the chromatic roots closer to the disk Ir - 11 5 1.

It was shown in [21] that CO-roBk is an upper bound for It - 11 where r is any chromatic

root of the graph. The roots of any subdivision of G, therefore, are bounded by 1

plus the co-rank of G. However, more is true: in [20] it was proven that for any E > 0,

t here is some subdivision of G having all its chromatic roots in lz - 11 < 1 + E. This

belies the empincal evidence that all large subdivisions have their chromatic roots in

the salient disc. Our results here are indeed strong enough to prove t his fact .

Theorem 2.2.11 For any E > O , there is an L = L(G, E ) such that, i j we svbdivide

each edge of G into a path of length at l e u t L, then ail chromatic mots of the resultzng

graph G lie in 1

Proof Let e - that we subdivi

> O be given. Suppose E = {el, ..., e,) are the edges of G, and

ide edge ei into a path of length l i , i = 1, ..., m. We obtain a

whose chromat ic polynornial, by Theorem 2.2.5, is T (G:,:;;;;_", x) =

is the expression (in braces) in Theorem 2.2.5 (with k replaced by m).

As we remarked eariier, the roots of r(G;,',_C"', x)? and therefore of are

bounded by the c+rank p of G. Let C = C(G, E ) > O be a bound for the maximum

modulus of the f's and g~ on 1 + E 5 II - z( 5 p. Choose L > O large enough that cn 1+€ > 2" - 1. Suppose that 4 , . . . , lm are all larger than L, and, without loss of C

generality, l1 5 l2 5 - - < 1,. Let z be such that 1 + E 11 - zl 5 p; we will show

JE ,,..*, lm(2)( > 0.

To that end, note that r (G - E, r ) = zn, whose modulus is at lest E". Set

y = 11 - zl. Then, by the triangle inequality,

On the right hand side of the above there are exactly 2m - 1 terms in y (without

combining any terms of like degree). We rewrite the expression as

where p = 2m - 1 and nl 2 nl 2 2 n,. In particular, nl = ~ ~ , l i and

n~ = CE2 l i . Then

which completes the proof. 0

Direct calculations with generalized theta graphs suggests that this may be only

half of the story. More specificdy, we conjecture that the region lz - 11 < 1 + E in

Theorem 2.2.11 can be replaced by {z E @ : 1 - E < lz - 11 < 1 + E ) u (1). Returning to the proof of the theorem, all we really needed of G - E was the fact

that it has no roots on 1 < lz - 11 < p, for then we knew it is bounded away £rom

zero on 1 + E 5 Ir - 11 5 p for any h e d E > O. So in fact any subset Et of E for

which every root of G - E' lies in )z - II 5 1 will do. Conversely, if E' is a subset of

edges such that G - Et has a root on Iz - 11 > 1, then by (ii) of Theorern 2.2.7 and

the remark immediately folIowing the theorem. there will certainly be an E > O and

an 1 siich that GY has a root on (z - 11 2 1 + E. Hence, in the statement of Theorem

2.2.11, we c m replace G by Et if and o d y if the graph G - E' has d its chromatic

roots in lz - 11 5 1.

Chapter 3

Roots of Independence

Polynomials

For a graph G with independence number B- let il denote the number of independent

sets of vertices of cardinality k in G (k = 0,1,. . . , P ) . Several papers exist (cf.

(2, 23, 34, 36, 411) on the independence sequence (il, il, . . . iP) of a graph (or its

cornplement), exploring various such problems. The independence polynomial of G,

is the generating polynomial for the sequence. The path P4 on 4 vertices, for example,

has one independent set of cardinality O (the empty set), four independent sets of

cardinality 1, and three independent sets of cardinality 2; its independence polynomial

is then ip,(x) = 1 + 42 + 3 9 .

As is the case with other graph polynomials, such as chromatic polynomials (cf.

[25, 51]), matching polynomiab ([38, 39]), and others, it is natural to consider the

nature and location of the roots. Interesting in their own right, they can shed some

light on the underlying combinatorics es weU. Newton (cf. [31]), for example, showed

that if a polynomial f (z) = a& with positive coefficients has all real roots,

then the sequence t ~ , a l , . . . , ag is log-concave (i.e., a: 5 ak-iak+i for a.li k), and

hence unimodal (for some k, al 5 = - C ak 2 ak+1 2 - - > a,). Unimodality

conjectures permeate combinatorics (se, e.g., [56]) , and it was conjectured in [23]

that the independence vector (zo, il, . . . , ip) of any well covered g a p h (cf. Section 3.1.1

of this thesis for the definition) is unimodal, and some partial results in that regard

have been obtained via the roots of independence polynomials (cf. [23]). Further

results on independence polynomiais and their roots can be found in [23, 35, 40, 421.

One iine of research in chrornatzc roots has been determinhg the topological cl*

5wes of both the real and complex roots of the set of all chromatic polynomials. It

was shown between the works of Jackson [45] and Thornassen [57] that the closure

of the set of real roots of chromatic polynomials is {O) U {l) U [32/27,m). Recent

work of Sokal (cf. [Zi]) shows that chromatic roots in general are dense in the entire

cornplex plane; even when restricted to plana graphs, the complex roots are dense in

@. except possibly in the disk Ir - 11 5 1. Our first order of business in this chapter

will be to answer t hese same questions for roots of independence polynomials.

Much of our work stems £rom the following key result. For two graphs G and H,

let G[H] be the graph with vertex set V(G) x V(H) and such that vertex ( a , x ) is

adjacent to vertex (6 , y) if and only if a is adjacent to b (in G) or a = b and x is

adjacent to y (in H). The graph G[H] is the lexicog~aphic product (or composition) of

G and H, and can be thought of as the graph arising from G and H by substituting

a copy of H for every vertex of G.

Theorem 3.0.12 Let G and H be graphs. Then the independence polynomial of

G[H] is

Z ~ H ] (x) = iC (iH (z) - 1). (34

Proof By definition, the polynomial i c ( i~ ( z ) - 1) is @en by -

where if is the number of independent sets of cardinality k in G (similady for i:).

Now, an independent set in G[H] of cardinality I mises by choosing an independent

set in G of cardinality k , for some k E {O, 1,. . . 1 ) , and then, within each associated

copy of H in G[H], choosing a nonempty independent set in H, in such a way that

the total number of vertices chosen is 1. But the number of ways of actually doing

this is exactly the coefficient of xL in (3.2), which completes the proof. 0

By applying (3.1) to the right families of graphs, we will be able to determine the

closures of real and complex independence roots. We will then move on to consider

the special case of higher compositions of a graph with itself, asking for where the

independence roots are approaching. Since the polynomials involved are essentially

higher compositions of ic(x) - 1 with itself, it may not be too surprising to leam that

the generic case is convergence to a fractal.

We shall have occasion to make use of an easy recursive formula for calculating

independence polynomials.

Proposition 3.0.13 ([23, 441) For any vertex u of a graph C,

ic(x) = ic-,(x) + x iC-[,,l (x).

where [v], the closed neighbourhood of u, consists of v, together with all vertices inci-

dent with v.

Proof For k 2 1, an independent set of k vertices in G either contains v or does not. - There are if? that do, and if-" that do not. Thus, for each k >_ 1, the coefficient

of is the same in both sides of the above equation; and both sides clearly have

constant term 1. The two polynomials are therefore equal. 0

Another useful tool is:

Proposition 3.0.14 For graphs G and H, the independence poiynomzd of Meir dis-

joZnt union G W H is imH(z) = iC(x) * iH(z)-

Proof For k > O, an independent set of k vertices in G ttl H arises by choosing an - independent set of j vertices in G (for some j E {O, 1, . . . , k)) and then an independent

set of k - j vertices in H. The number of ways of doing this over a l l j = 0,1,. . . k

is exactly the coefficient of xk in ic(z) - i H ( x ) . Hence, since both sides of the above

equation have the same coefficients, t hey are identical polynomials. 0

3.1 Location of Independence Roots of some Fam-

ilies of Graphs

As advertised, we shall now find the topological closures of real and cornplex inde-

pendence roots. As the coefficients of any independence polynomial are positive al1

the way down to the constant term? it is clear that no real independence root is

nonnegat ive. Nevert heless, we have:

Theorem 3.1.1 Cornplex roots of independence polynomiuls are dense in al1 of @,

while real independence mots are dense in (-cm, O].

We will prove the theorem by considering very specific families of graphs, taking

t heir lexicographic product , and examining the roots of the independence polynomials

which arise. The upshot will be the truth of the Theorem 3.1.1 even for some very

restricted f d e s of graphs, n d y well covered and comparability graphs.

3.1.1 Well Covered Graphs

A graph is well couered if every maxi.mil set of independent vertices has the same

cardinality. The graph Cd, for instance, is welI covered with independence number 2,

while Cs, a gaph with independence number 3, is well covered, since it contains

maximal independent subsets of cardinality 2. Well covered graphs have attracted

considerable attention; see [49] for an extensive survey.

Well covered gaphs are closed under lexicographic product .

Proposition 3.1.2 If G and H are well couered, then G[H] is aho well covered.

Proof Any independent set in G[H] arises by choosing an independent set W in - G, and for each vertex in W , an independent set of vertices X in H. Every maximal

independent set in G[H] will therefore have cardinality &&, and so G[H] is well

covered . O

Denote by [lJ] the set (1,2,. . . , P l , and (as in [23]) L! (where k is a positive

integer) the graph with vertex set [l, BIk, in which two k-tuples are joined by an edge

if and only if they agree in a coordinate.

Proposition 3.1.3 For 0, k 2 2, the graph L i is well covered with independence

number p.

Proof Let {vl , y, . . . , v,) be an independent set of s < vertices. Then, for each - i = 1,2, . . . , k there is a number x(') E (1, ,O] which is not equal to the i-th coordinate

of any V j ( j = 1,2,. . . , s). Thus, the vertex u,+l = (dl) ,x(*), . . . ,x(") is not adjacent

to any vj (j = 1 ,2 , . . . , s), and so {ul, ~ 2 , . . . vs+l} is not a maximal independent set.

Hence, any maximal independent set has cardinality at least B. On the other hand, let {v1,î)2,. . .us} be a set of s > vertices. Then two of

them must agree in the hst coordinate, by the Pigeonhole Principle. Therefore, no

independent set has cardinality greater than P. This completes the proot a

The graphs Li were considered in [23], where the following was established.

Theorern 3.1.4 ([23]) If 2k-1 2 8 $: 1 then the srnallest zen, y f ) of i, (z) lies in B

the znterval

-p < $' < -P(1- 2-k) .

By taking the lexicographie product of the Li with complete qaphs, we fmd be-

low that among the independence roots which arise are red roots that are dense in

(-a, O]. Complete graphs are obviously weU-covered (wit h independence number 1 ) , and iKn(x) = 1 +m. Proposition 3.1.2 then implies that L$[K,] is weu-covered, and.

by equation (3.1), z ~ $ [ ~ ~ ] ( x ) = +(nx).

Theorem 3.1.5 The real independence roots of the family {L$[KnJ} are dense in

(-m, O]

Proof Note that iLiLlKnJ (x) = iL# + nx - 1) = ÊL$m). Let s (-m, O] and - E > O be given. Begin by choosing a positive integer n large enough that the intenml

n - (s - E , S + E ) = (ns - w , n s + n ~ ) contains some integer 5 -2. Next, from

Theorem 3.1.4, we can choose a k such that ip(x) has a root r in that interval. Then

r /n E (s - E , s + E ) ? and

completing the proot

It turns out that complex independence roots of wel1 covered graphs are dense in

all of @. To prove this, we will compose the graphs L$[K,,] with empty graphs FB2. The empty graph is obviously well covered (with independence number n). and

(from Proposition 3.0.14) +(x) = (1 + x)".

Theorem 3.1.6 The zndependence nwts of the family L$[K,,] [z,] are dense in @.

Proof Set R = {real roots of the family L$[K,,] [G]); fiom Theorem 3.1.5 we - know that R = (-oc, O]. Let r E C and E > O. We will show that there exist positive

integers p, k, nl, nz such that iLi[Knl , (K~~, (2) = O for some Z within an E-radius of z.

Rom Proposition 3.1, we have

We may assume z # -1; thus Ir + il > O. Choose nl large enough that some r(2k+l)r

nz-th root of - l z + iIn2, Say w = Iz + ile '2 , is within an ?radius of z + 1.

Choose 6 > O such that 6 < 5 and r = -(Ir + 11 + 6)n2 - 1 E R (by continuity, such

a 6 exists). Then the corresponding n2-th root of r + 1 = -(Ir + II + 6)"1, namely i(Zk+l)r

G = ( I z + 11 + 6)e 2 , is within an &-radius of z + 1, as

Finally, since r E R, there are numbers 8, k, ni such that Z , ~ ~ ~ , , ~ ( T ) = O. Set i =

G - 1. Then

li - zl = I(W - 1) - z ( = 12U - ( 2 + 1)1 < E ,

and

complet ing the proof. O

Theorems 3.1.5 and 3.1.6 imply that Theorem 3.1.1 is true even when restricted

to weil covered graphs.

3.1.2 Comparability Graphs

A simple graph G is a cmparability graph if it has a transitive onentution, that is,

an orientation of its edges such that if x + y and y -, z then x -. z. Comparability

graphs are also closed under graph composition.

Proposition 3.1.7 If G and H are comparabzlity graphs, then G[H] is also a corn-

pambzlity graph.

Proof Orient the graph G[H] by ( a , x ) < (b, y) if and only if a < b (in G) or a = b

and x < y (in H ) . This is a transitive orientation of G[H]. For suppose (a, x) < (b, y)

and (b, y) < ( c , ~ ) . If a = b = c, then x < y and y < z, and so x < z by transitivity

of H, whence ( a , x ) < ( c , t ) . If instead a = b < c or a < b = c or a < b < c, then

a < c by transitivity of G, and so (a, x) < (c, z ) . This completes the proof, as there

are no other possibilities.

Contained in the collection of comparability gaphs are paths, complete graphs,

and empty graphs.

Proposition 3.1.8 Paths, complete graphs, and ernpty graphs arc! al1 cornparubility

graphs.

Proof Going from left to right through P,, the path on n vertices. orient the first

edge forward, the second backward, third forward, and so on. This gives a transitive

orientation (trividy) , and t herefore P,, is a comparability graph.

To see that Kn is a comparabiiity g a p h , simply label its vertices 1,2, . . . , n, and

orient edge zj a s i -, j if and only if i < j . The transitivity of this orientation follows

that of < on IR. Findy, Zn is trivially a comparability graph. O

Together with Proposition 3.1.7, this irnplies:

Corollary 3.1.9 The graphs P,, [K,,] and P,, [K,,] [K,,] are cornparabdi$ g7aphs.

Analogous to what we did for well covered gaphs, we will show that the family

{Pn, [K,,] ) has real independence roots which are dense in (-00, O], while the complex

independence roots of the family {Pn,[K,1][K,3]) are dense in all of @.

We s t a t t with paths, themselves. The graph f i , for example, has independence

polynomial i f i (x) = 1 + 5x + 6x2 + 2. Its roots are shown in Figure 3.1; they are ail

real. We can Say much more:

Figure 3.1: The independence roots of P5.

Theorem 3.1.10 The independence roots of the famiiy {P') are real and dense in

(-OO, -a]. Proof Since P, is the line graph of Pnci, M(Pn+I, x) = xnip,(-i/z2), the former - being the matching polynomial of P,+l, and matching polynomials are known [42] to

have ody real roots. It follows that ip , (x) has only real roots as well. The reduction

in Proposition 3.0.13 for calculat ing independence polynomials gives

and so the family {i (x) ) is recursive; the initial conditions are ip, (x) = 1 + x and

iR (x) = 1 + 22. Solving, we find

w here

and dz (1 + 2x1

~~(x),Qz(x) = 2 J1+42 The non-degeneracy conditions of the Beraha-Kahane-Weiss theorem (Theorem

2.2.3) are therefore satisfied, and part (i) of that theorern Mplies that among the

lirnits of roots are those z for which

which simplifies to

I I + JïTZ1= 11 - JTTZl,

implying that Jw is purely imaginary. Thus 1 + 42 5 O, i.e., z 5 -114, which is

what we wanted to show. 0

By composing with complete graphs, we can fill up the rest of the negative real

a i s .

Theorem 3.1.11 The independence mots of the famdy P,,[K,,] are real and dense

in (-m, O].

Proof From the previous theorem, independence roots of the graphs P,,[Kl] = P,, - are real and dense in (-a, 4 / 4 1 . Now let s E (-1/4,0) and E > O be given. Then

there are positive integers nl and n* for which iPni [ ~ ~ ~ i (x) = iPnl (n2x) has a root in

(s - a, s + E ) . For choose nz large enough that n,s 5 -114. Then, £rom the previous

theorem, we can choose a number ni such that ipml (2) has a root T nz (s - E, s + E ) . But then & r E (s - E , s + E ) and iPn,IK,l(q r ) = ipn1(n2 q) = iP,,(r) = 0,

completing the proof. a

Compositions with empty graphs will then fill up the complex plane.

Theorem 3.1.12 The independence mots of the graphs P,, [K,,][K,] are dense in

Proof The proof is essentidy the same as that of Theorem 3.1.6: starting with - any collection of graphs whose independence roots are dense in (-oo, O], compositions

with empty gaphs yield independence roots which are dense in @. 0

Hence, Theorem 3.1.1 remains true when we restrict to comparability graphs.

3.1.3 On Further Classes of Graphs

It may be of interest to study the independence roots of yet hirther classes of graphs.

Some common ones include chordal, interval. claw-free. and line graphs:

A simple graph G is a chordal graph if every cycle in G of length at Least 4 has

a chord.

Given a family of intervals. we can define a graph whose vertices are the inter-

vals, with vertices adjacent when the intervals intersect. A graph formed in this

way is an intemal graph, and the family of intervals is an intemal representation

of the graph.

A simple graph G is claw-free if does not contain a s an induced subgaph.

The fine graph of a graph H, is the graph G = L ( H ) whose vertices are the

edges of C, with e f E E(L(G)) when e and j share a vertex.

It can be shown (cf. [59]) that interval graphs are chordal, and line graphs are

claw-free. The reader can venfy that the graphs Pn,[Kn2] are chordal, intenml, and

claw-free, while graphs P,, [K,,][K.,] (ns 2 2) are neither; paths Pn are line graphs,

while graphs Pn,[Kn,] are not. We sumrnarize our results in the table below:

-

Class I Real Roots

1 Limits Generating Family

Complex Roots

Limits Gen. Family

* fu t her investigation necesssry

3.2 The Independence Attractor of a Graph

WeU Covered

Comparability

Chordal

Interval

Claw-Free

Line

Since lexicographic product is associative, we may speak of powers G~ = G[G[G . -11 - k

of a gaph G without ambiguity (G1 = G). For G = 6 (a path on three vertices) the

--

(-00, O] Lg [Knl

(-w , O] pn1 w . 2 1 (-00, O] pn, [Km1

(--ml 01 PL K 2 1 (-00, O] P ~ I K t 2 1

(-00, -$* Pn

independence roots of G7 are shown in Figure 3.2. It appears that the independence

roots of Gk are approaching a fractal-like object as k -. m. We are lead to ask:

Question 3.2.1 For a gmph G , whut bappens to the mots of the independence poly-

nomials ÉCk (x) as k --+ a, ?

In section 3.2.2 we are able to describe precisely where the roots are approaching,

and in what sense they do so. We then ask the question of when these iimiting

sets are connected. In section 3.2.3, we provide a complete a m e r for graphs with

independence number 2. We then a m e r the question for some idbi te families of

graphs having arbitrarily high independence numbers, in section 3.2.4.

Some familiarity with iteration theory will precede any reasonable understanding

of this material. With that in mind, we c o k t in the next section the relevant nota-

tion, terminology and results from the field, dong with some references. Incidentdy,

while Theorem 3.2.10 will have most direct application for us, it cannot (as fax as the

author is aware) be found explicitly in the literature.

Figure 3.2: The independence roots of G7, where G = f i .

3.2.1 Julia Sets and the Iteration of Rational F'unctions

Except where otherwise stated, any definition or assertion made in this section can

be found in Beardon's book [7]. Much of the information can also be found among

the works of Blanchard [15] and Brolin [19].

In section 1.2.3, we introduced the space (Cm, no) - the extended cornplex plane

C,, with the spherical metric 00. A rational map on C, is a hinction f ( z ) =

p(z) /p(z), where p(z) and q(z) are polynomials; its degree is max{deg p(r ) , deg q(2)).

It is weil known that each rational map is analytic throughout @, (neither w nor

any poles of f are singularities).

Rational maps of the form

are cded M6bius maps; the condition ad - bc # O ensures that 4 is one-teone and

thus invertible. Two rational maps f and g are wnjvgate if there is some Miibius

map 6 such that

= 6 f a(-') where $O(-') is the inverse of 4. It is then easily veriiied that, for any k,

an important property of conjugacy. By fO*, where k 3 1, we mean f 0 f 0 * 0 f if -- T

k

k 2 1. Also, define f "(O) to be the identity map.

Associated with each rational rnap is a set (most often, a Eractal) known as its

Julia set.

Definition 3.2.2 A family 3 of maps of @, into ilselfis eqvicontinuous ut zo zf and

only if for every E there erists a positive 6 such that for ail z E C,, and for al1 f in

3,

2) < 6 zmplies a( f (zo), f ( z ) ) < E.

The farnily F is equicontinuow on a subset X of @, i f it is equicontinuous ut each

point zo of X .

Roughly speaking, equicontinuity means preservation of proxirnity. Unless other-

wise stated, f is a rational map from @, into itself.

Definition 3.2.3 The Fatou set F ( f ) is the mazimd open svbset of C, on which

the fumzly { f is equicontznuous, and the Julia set J( f) is its complernent in

The Julia set of f (z) = 3x3 + 9x2 + 72 is shown (in black) in Figure 3.3. A method

for computing such pictures is suggested by Theorem 3.2.7 below. The actual Maple

procedure used to generate the pictures of Julia sets found in this thesis is provided

in the Appendix.

It is not hard to prove the following.

Figure 3.3: The Julia set, J(3x3 + 9x2 + 72)

Theorem 3.2.4 If g is a rational map which is conjugate to j - Say g = 4 O f O

for sorne Mobius map 4 - then F ( g ) = $(F(f)) and J ( g ) = #(J(f)). The sets

J ( g ) and J( f ) are then said to be analytzcally wnjugate, as are F ( g ) and F( f).

Furthemore, F (f = F( f) and J ( f O&) = J( f) for any positive integer k .

By definition, F( f ) is open, while J( f ) is compact. If deg( f ) 2 2, then J( f) is infinite; in fact, J(f) is a peifeet set (and hence uncountably infinite by Baire's

Category Theorem), that is, J = Ac(J). The sets F and J are each wmpletely

invanant under f , in that f (F) = F = f " ( - ' ) ( ~ ) and f (J) = J = / O ( - ' ) ( J ) .

Periodic points of rational maps play an important role in iteration theory. A

point q is a periodic point of f if, for some positive integer k, f 0k = 1. The

smdest positive integer k for which f O k ( % ) = q is the period of a. The forward

orbit O+(*) = {fok(zo)) of a periodic point zo is cded a periodic cycle. Periodic

points of period 1 are the 6xed points of f .

For a periodic point zo E C of period k, the number X = (f "')' (zo) is the multiplier

of its periodic cycle, and is independent of the choice of zo from the cycle (a simple

exercise involving the Chain Rule). The cycle is

(i) attractzng if O < IXI < 1 (and super-attmctzng if X = 0);

(ii) repelling if IX 1 > 1;

(iii) ratzonally indzfferent if X is a root of unity; and

(iv) irmtionally indifferent if JX/ = 1, but X is not a root of unity.

Theorem 3.2.5 Repellzng periodzc cycles all lie on J ( f ) , and in fact are dense on

J ( f) . Ratzonally zndgerent cycles also l ie on J ( f). Attrizcting cycles, on the other

hand, lie in F( f ) . Meanuhile, an irationally zndzffe~ent periodic cycle may lie in

F(f) or it may lie on J(f).

If zo is an irrationdly indifferent periodic point of period k, lying in F ( / ) , then

the cornponent (i.e., maximal open connected subset) Fo of F( f ) containing zo is

forward invariant under /Ok, and is cdled a Siegel disk. For any point z # zo in Fo,

the sequence { f ~ ~ ( z ) , fa(*)(=), . . .) is dense on a curve - called an invariant circle -

lying inside Fo.

In fact, a complete classification of the possibilities for periodic components of a

rational map is known; and e u q component C of a Fatou set F( f ) is eventudy

periodic under f , that is, for some j > k 2 O, ~ J ( C ) = PC(C). These very deep

and fascinating results were proved by Sullivan (cf. [7] for references and detaüs). An

immediate consequence of his work is:

Theorem 3.2.6 If f is a polynornial, and E F ( f ) , then the fonuard orbzt O+(*) =

{f"'(zo)) either

(i) converges to a periodic cycle, or

(ü) settles znto a 'periodzc cycle' of Siegel disks, 6ecumzng dense on an invariant

circle in each.

Now, the backward orbit of a point z E @, is the set

O- (2) = ugo f (z) .

We are using the notation f O ( - ' ) for the set-valued inverse of f (a function on subsets

of C,), and f"(-" = f'(-') O f O(-') o O f "(-'1. If 0-(2) happens to be finite, t hen \ I

k z is called exceptional. For example, if f (x) = xn, then both O and oo are exceptional

points off . A rational rnap f has at most two exceptional points, and they necessarily

belong to F( f ) . The point w is an exceptional point for every polynomial p; thus,

oo E F ( p ) and, since J ( p ) is closed, J ( p ) will be bounded as a subset of (C, 1 . I), where 1 - 1 is the Euclidean metric on C, discussed in section 1.2.3. Backward orbits

are closely linkecl to the Julia set:

Theorem 3.2.7 Let f be a rational map with deg( f ) 2 2.

If t is not exceptional, then J ( f ) E Cl (Om(r)).

Since the inverse images f O( -~ ) (Z~ ) are finite, they are necessarily compact. Instead

of looking at the entire inverse orbit O-'(zo), we could ask whether the sets f "(-')(zo)

wiU actudly converge to J ( f ) under the Hausdorff metric h, on 'H(C,) (cf. section

1.2.3). As it tums out, the generic case is convergence to J( f ). While a result of this

nature cannot be found explicitly in the literature, however, there are enough tools

adable to piece one together. Our efforts in this regard will pay off immensely in

the sections that follow. One tool we will need is:

Theorem 3.2.8 (cf. (71, p. 71) Let f k a mtional map of degree at le& two, and

E a compact svbset of @, such that for al1 z E F(f) , the sequence { f ~ ~ ( z ) ) does not

accumulate at any point of E . Then for any open set U containing J ( f ) , f O(-~) (E) Ç

U for all sufic2ently large k.

Another is:

Theorem 3.2.9 (cf. [7], p. 149) Let f be a rational map of degree ut leust two, W

a domain that meets J , and K any compact set containing no exceptional points of

f . Then for all suficzently large k , f o ~ ( W ) > K .

(A domain is an open connecteci set). Together with Theorem 3.2.6, these are strong

enough to prove:

Theorem 3.2.10 Let f be a polynomial, and zo a point which does not lie in any

attracting cycle or Siegel disk off . Then

Proof' Let E > O be given. Establishing the above limit is equivalent (cf. 161. p. 35)

to proving that, for al1 sufficiently large k,

and

(i i) J ( / ) C f "(-*) (z0) + E.

where A + E = { z : go(z, a) < E for some a E A), the "dilation of A by a bal1 of

radius E" ([6], p. 35).

To prove (i), note first that if zo E J(f), then fO(-')(a) C J(f) C J(f) + e for

all k. If instead zo E F ( f ) , then, since we are assuming that zo lies in neither an

attracting cycle nor a Siegel disk, Theorem 3.2.6 impiies that no point z in F( f )

will have a forward orbit which accumulates at (recall hom Theorem 3.2.5 and

the remarks immediately foilowing the theorem imply that any non-attractive (and

thus irrationally indifferent) periodîc cycle in F( f ) necessarily lies in a Siegel disk).

Hence, the set E = {zo) satisfies the hypothesis of Theorem 3.2.8, and therefore

f"-')(zo) E J ( f ) + E fm ail sufliciently large k.

'The author is indebted to Professon John Milnor and Robert Devaney for sharing with him their own thoughts on the possible correctness of this result, at a time when the author had not yet completed his proof.

To prove (ii), we begin by choosing a positive number b < 4 2 , and covering

J( f ) with finitely many open balls of radius b (such a covering exists since J( j) is

compact). The point q is not exceptional, since exceptional points are necessarily

periodic points in F( f). For each ball W in the covering W, Theorem 3.2.9 implies

that, for al1 sufficiently large k, fOk(W) > {zo) , and hence f "(-&)(zo) n W # 0. Since

there are only finitely many such balls, we then have that, for ail sufficiently large k,

f O(-~)(Z~) n W # 0 for each ball W. Finally, since E > 26, it follows that, for all such

k, / o ( - ~ ) ( ~ o ) + E > W > J(f). O

3.2.2 Independence Attractors of Graphs: a General Theory

We set out now to describe just where the independence roots of powers CL of a graph

G are 'approaching' as k -r cm. The upshot will be an association of a fractal with a

graph. For each k 2 1 the set, Roots (ict), of roots of iGk is a finite - and therefore,

compact - subset of (C, 1 1). We ask whether the limit of the sequence {Roots (ic*)}

with respect to the Hausdorff metric hl., on X(@) (cf. section 1.2.3) exists in general.

In fact, it does.

Definition 3.2.11 The independence uttractor of a g~aph G is the set

where the fimit is taken in ( l i (C) , hiSI).

That Z(G) actually d t s for every gaph G is part of Theorem 3.2.14 below, the

main result of this section. Let us begin with a simple but important characterization

of the right hand side of equation (3.4). For each k 2 2, associativity of graph

composition allows us to write Gk = G k - l m , and Proposition 3.1 then implies that

which in turn leads to the relation

Roots (ic.) = (ic - 1)"(-') (Roots (ic.-t)) .

Also,

Hence,

Proposition 3.2.12 Setting fG(x) iC(x) - 1, we have

4-4 Roots (ic.) = fG (-1)

for each k 2 1. Therejo~e,

With this observation, we axe almost in a position to apply the results of the

previous section to the problem of describing Z(G), but one point remains to be

addressed. We are attempting to take the lirnit (3.5) in (31(@). hl.,), as opposed to

('H (Cm ) , h, ) . However, note the following:

Lemma 3.2.13 The sepence { f~(-~)( - l )} is bovnded in (@, 1 . 1).

Proof The result will be obvious if we can prove the existence of a number R > O

such that Ir1 > R implies 1 f&)J > R. To that end, for f&) = xfZi ikxk, let

C = max &,andset R = m a x - i ~ k s p - i

( {z, 5 + 1 ) By the trian jie inequality,

which is what we wanted to show.

Applying Theorem 1.2.7, we may instead consider the limit (3.5) as being taken

in (%(Cm), h,), which means that we can apply the techniques and results of the

previous section on iteration t heory.

Theorem 3.2.14 answers completely Our question in general, and is the foundation -

for everything that foilows. Empty graphs will not be of interest, for if G = K,, then -

Gk = K,,t and ic, (x) = (1 + z ) "~ , whence Z(G) = { - 1).

Theorem 3.2.14 Let G be a non-empty graph, and denote by q(G) the multiplicity

of -1 as a mo t of ic. Set fc = ic - 1.

(i) If i)(G) 5 1, then

z(G) = J ( fc). the J d i a set of fc.

(ii) If q(G) 2 1, then

- 0(-~)(-1)) . Z(G) = Cl (uksi Roots (ick)) = Cl (ukZi fc

In case (ii), iGk is divisible by (ic~-t)q(G) /or each k 2 2, and

iim (Roots (iok) \ Roots (iG.-1)) = lim Roots ( ickiol) = J(fG/. (3.6) &+a k d o o (iGk-1)

Further, for q(G) > 1,Z(G) fS partitioned by the set, Uk>i - Roots (ick), and its accu-

rnulation points, J( fc) .

Proof If G has independence number 1, then G = K, for some n 2 2, and - ic(x) = 1 + m. It foUows easily that for each k 2 1, ici;(x) = 1 + nkx, whose only

root is -l/nk, which tends to O as k tends to infinity. Thus, Z(G) = {O) for graphs

G with independence number 1. As zc(x) = 1 + nz where n 3 1, we have q(G) = 0;

also, J(nx) = (O) as the forward orbit of any non-zero point will go off to infinity.

This proves the resdt for graphs with independence number 1.

Let G be a non-empty graph whose independence number is at least 2. For the

cases q(G) = O and q(G) = 1, we will show that the hypothesis of Theorem 3.2.10 is

satisfied for f = fc and zo = -1. To that end, note first that since fc has integer

coefficients, the forward orbit { fzk(- 1)) of - 1 lies entirely in Z, and therefore could

not possibly be dense on any invariant circle in a Siegel disk.

Consider first the case q(G) = 0: here, f&l) # -1 as fc(-l)+l = ic(-1) # 0.

We wiil show that in fact -1 cannot be a periodic point of fc of any order. Together

with our observation above, this will show that the hypothesis of Theorem 3.2.10 is

satisfied here. We argue by contradiction. Suppose fzk(-1) = -1 for some k 2 2. o(k-1)

Then fc(fc (-1)) + 1 = 0, and so f~'~-')(-l) is an integer mot of fc + 1, a - €2

polynomial having positive integer coefficients and constant term 1. Applying the O(&- 1) Rational Root Theorem. it follows easily that f, (-1) = -1. Repeating this

argument, we conclude that fc(-1) = -1, a contradiction. Therefore, - 1 is not a

periodic point of fc, and the hypothesis of Theorem 3.2.10 is satisfîed.

Next, consider the case q(G) = 1: here, fc(-1) = -1 and f&(-1) = (fc + 1)'(-1) = ib(-1) # O. Since f& has integer coefficients, this implies that If&(-1) 1 2 1. If If&(-1) 1 = 1, then f&(-1) is either +l or -1, implying that -1 is a rationally

indifferent fixed point of fc. If instead 1 f&(-i)l > 1, then -1 is a repelling fixed

point of J(fc). In either case, the point zo = -1 belongs to J( fc), and once again

the hypothesis of Theorem 3.2.10 is satisfied. In addition, as -1 E J(jo), it follows

from Theorem 3.2.7 that J(f') = Cl (ukZl fëk(-1)).

When q(G) > 1, fc(-1) = -1 and f&(-1) = (fc+ 1)'(-1) = z&(-1) = 0; thus,

q = -1 is an attractzng fixed point of fc, and so Theorem 3.2.10 does not apply. Let

E > O be given. We need to show that

for ail sufEicient ly large k.

Part (a) is obvious, as f$')(-1) C ukZi j$')(-l) for al1 k. Part (b) is not

difEcult either: Since f (-1) = -1, we have -1 E fi(-"(-1). Applying f l ( - ' ) to both 4-11 a(-2) o(-(k-N (- sides, we find that fc (- 1) C fc ( -1) . By induction, then, j, -

4-'1 fc (- 1) for each k 2 2. Consider the family of open sets { 1 ) + E ) . Rom

what we have just shown, this forms an zncreasing open cover of the compact set

Cl ( ~ k > ~ - f ; ( ( - ~ ~ ) ( - 1 ) ) , and hence the latter will be contained in { f$k)( - 1 ) + E ) for

al1 sdficiently large k, which is what we wanted to show.

Now we prove (3.6). Let G be a graph such that v(G) 2 1. Then -1 is a root of

ic Let R denote the set of mots of ic; note that R = f$')(-1). AS G is non-empty,

ic(s) is not of the form (1 + z)@, and so fi = R \ {- 1 ) is non-empty. Consider any

element r of E. As j&) = - 1, fgk(k(r) E Z for al1 k 2 1. Thus, r cannot be a periodic

point of fc, for if it were, then r E Z. However, since zc(s) has positive coefficients

and constant t e m 1, its only integer root, by the Rational Root Theorem, is -1.

And s o r = -1, contradicting the fact that r ER. Now. as f ~ ~ ( r ) E Z for all k > 1,

the sequence {fzk(r)) will not be dense on any c w e , and so r can lie in no Siegel

disk either. Hence, the hypothesis of Theorem 3.2.10 is satisfied for each member r n,

of the finite set R, and so

Write i&) = ( i + ~ ) " ~ ) & ( z ) , where &(+) has mots a. Then i @ ( x ) = iG( f~(x)) =

(iC(z))q(G' @G(fC(~)). By induction, vue find

and t herefore

k 1 4 - j ) " "'-'*-'"(R) = uj% fG ( R ) U {-1) made in the second 1 s t equaiity is The claim fc 4 - 1 1 seen by a simple inductive argument, noting that R = R ü (-1) and fc (- 1) =

R U {-il. To see why the 1st equality (above) holds, suppose that r, s ER were such that

f;(-(k-l)) ( r ) n 12-"(s) # 0 for some j < k - 1. Then r = f "('-l-j)(s) = - 1, n d

contradicting the assumption that r ER. This completes the proof of (3.6). O(-k) " Findy, since hmk,, fG ( R ) = J ( fc), and J ( fG) is a perfect set (i.e., equal

will accumulate to no less than to its accumulation points), the union fc J( fc) It can accumulate to no more than J ( fc) either, since any accumulation point

necessarily lies in the lirniting set. Hence,

And for q(G) > 1, J ( fc) is disjoint £rom ~ ~ ~ & ~ ) ( - 1 ) , as -1, being an attractive

fixed point of fc, does not belong to J( f G ) This conchdes the proof of the theorem.

O

Early into the proof, we observed that the independence attractor of any graph

with independence number 1 is just (-1). Thing get more interesthg as we consider

paphs with higher independence number, and we will sometimes find it convenient

to work with a polynomial gc to which fc is conjugate.

Theorem 3.2.15 If, for a graph G, a Miibius transformation # and poïyzornial gc

are svch that gc = # O fc O bO(-'), then

(i) fc f ies -1 if and only if gc &es 4(- l),

(ii) the multiplicity of - 1 as a mot of f&) + 1 equals the multiplicity of 4(- 1 ) as

a mot of g&) - #(-1), and

(iii) I (G) = J(fc) zf and only if Z(G) = @ ' ) ( J ( g c ) ) .

In short, #(-1) is to gc what -1 is to fc.

Proof To see (i), observe that gc(#(-1)) = #( fc(@(-l)(-l))) = #( fc(-1)), which

(since 4 is injective) is equal to #(- 1) if and only if fc(- 1) = - 1.

To see why (ii) holds, just note that g&) = #(-1) 4(jc(@'(-')(r))) =

@(-1) - fc(#~(-l)(z)) = -1.

Finally, property (iii) follows immediately from the conjugacy property of Julia

sets, mentioned in Theorem 3.2.4. Specificaliy, since gc = 4 O fc O @l), J(gc) =

$ ( J ( fc)). Applying qY(-') g i m J( fc) = $"(-l)(J(gG)). O

As Julia sets are typically fractals, we are in essence associating a fiactal Z(G)

with a graph G. The question arises as to the possible connections between the two

objects. How are gaph-theoretic properties encoded in the fractals? What does Z(G)

Say about G itself? Figure 3.4 shows the independence attractor of C = 5K2, the

disjoint union of five K2'5 Since i c (x) = (1 + Z X ) ~ , I ( G ) = J((l+ 2 ~ ) ~ - 1). The

attractor appears disconnecteci, in contrast to Z(P3) (cf. Figure 3.2). We ask here:

Question 3.2.16 For which graphs G is Z(G) cunnected?

We can give a complete answer for graphs with independence number 2.

3.2.3 Graphs With Independence Number 2

Let G be a non-empty graph with independence number 2, having n vertices and m

non-edges (i-e., G has exactly m edges) . (Recall t hat ernpty graphs have independence

Figure 3.4: The independence attractor, 2(5K2) = J ((1 + 2 ~ ) ~ - 1).

attractor {- l), and are therefore not of interest). Then

2 fc(x) = ic(x) - 1 = mz +nz.

As G is non-empty, ic(z) # (1 + 2)* and so -1 will h le multiplicity no greater

than 1 as a root of ic(x) . Applying Theorem 3.2.14, we have:

Theorem 3.2.17 If G is a non-empty gmph with independence number 2, then

W ) = J(f0).

The Mandelbrot set M is the set of all complex numbers c for which the Julia

set of the polynomid x2 + c is connecteci. For any other value of c, J (x2 + c) is not

only disconnected, but totdly disconnected (cf. [32], p. 246). Julia sets of this type

are often calleci fiactal dwt. A plot of the Mandelbrot set (a subset of the complex

c-plane) is shown in Figure 3.5. A well known fact (cf. [32]) is that M is containeci

in the disk Ici < 2, .

-2 ' Figure 3.5: The Mandelbrot set.

Let us then work with a polynomial of the form x2 + c to which fc(x) is conjugate.

It is straightforward to check that

where

and

For future reference,

2 Notice that the constant term c = 5 - (4) in gc(x) is independent of m. This means

that the comectivity of T(G) will depend only on how many vertices G has; the fact

that G is non-empty implies that n 2 3. The location of Z(G), though, depends on

both the numbers of vertices and edges in G, as Theorems 3.2.19, 3.2.20, and 3.2.21

below imply the following:

Theorem 3.2.18 If G zs a non-empty graph with independence number 2 hauing n

vertices and m non-edges, and z E I ( G ) , then

n (i) -- < 'Re (2) < 0, and m

fi (ii) Zm (2) = O, vnless n = 3, in which case -- 4 s Z m ( z ) -. 2m 2m

Graphs for which = 2 , n = 3

There are exactly two graphs with independence number 2 on n = 3 vertices.

nameiy KI tri K2, the disjoint union of a point and an edge, and 6, the path on

three vertices. Their independence polynomials are iKlriK? (z) = 2x2 + 3 1 + 1 and

LFI(x) = x2 + 32 + 1; thus, Z(KI & K2) = J(2z2 + 32) and Z(P3) = J(z2 + 32). For either graph G, equation (3.8) says that fc(x) is conjugate to the polynomiai

gc(x) = x2 - a. For G = Ki K2, equation (3.9) tells us that 4(x) = 22 + and 3 so #O(-~)(X) = +I - a, while for G = P3, 4 ( ~ ) = x + $ and c(-~)(x) = IX - - 2 -

Since -a lies in the Mandelbrot set, J(z2 - a) is connected. By Theorern 3.2.15,

1(G) = @ - ' ) J ( X * - i), and since, for either graph G, is a mere scaling and

shifting, l ( G ) must also be connected.

With a little work, one can determine a box containing J(x2 - i). The aut hor can

prove that for g(z) = z2 - and t E C, if either 1% (z)l > $ or Jlm (z)l > 2, then

I$ ( z ) 1 -+ oc as t + oc. Surely this is known, so we will not clutter this section with

the details. The box [- #, 41 x [-2, $1 containing J(g (z) ) is in fact best possible, as

the points I# and iqi d lie in J ( g ( 2 ) ) : the point z = $ is a repelling h e d point

of g, and t~"~(f $i) = &$) = # E J ( g ( z ) ) . Applying #O(-') to the box will give a

tight box containhg Z(G). We have proved:

Theorem 3.2.19 If G is o graph with independence number 2 on n = 3 vertices,

where either

(i) G = K1 w KZ and f(-')(x) = $x - $, or

The attractor, Z(G), is connected, the~efore, and

(i) Z(G = K I i+i K2) = J(2x2 + 3 x ) [-&O] x [-$,+], while

(ii) Z(G = P3) = J ( z 2 + 32) 2 [-3,0] x (-2, $1.

Figure 3.6: The independence attractor, Z(KI H K2) .

Plots of Z(Ki tti K2) and I ( P 3 ) are shown in Figures 3.6 and 3.7, respectively.

That they appear to have the same 'shape' agrees with the fact that each is just a

shifting and scaling of J (x2 - 4).

Figure 3.7: The independence attractor, Z ( f i ) .

Graphs for which P = 2 , n = 4

For a graph G with independence number 2 on n = 4 vertices (and rn non-

edges), equations (3.8) and (3.9) teil us that fc(x) is conjugate to gc(x) = $ - 2 via

g(x ) = mz + 2, that is, gc = $ 0 fc O @-'). Now J(x2 - 2) is weU-knom (cf. [32], p.

226) to be the interval [-2,2]; applying the map #f(-')(x) = ' x - 2 m to this interval

gives

Theorem 3-2-20 If G is a graph with independence number 2 hauing n = 4 vertices

and m non-edges, then

The graph G = K4 - e has

m = 1 non-edge. Then fc(z) =

as shown in Figure 3.8.

L A

independence number 13 = 2, n = 4 vertices, and

9 + 42 and, from Theorem 3.2.20, S(G) = [-4,01,

Figure 3.8: The independence attractor, Z(K4 - e ) = [-4, O].

Graphs for which P = 2 , n 2 5

If G is a graph with independence number 2 on n 2 5 vertices, then c = 7212 - ( 7 ~ 1 2 ) ~ < -2, which lies outside the Mandelbrot set. This implies that J(x2 + c), and

hence Z(C) = (3(x2 + c))), a mere scaling and shifting (by equation (3.10)) of

J(x2 + c), is kactal dust. In k t , we have:

Theorem 3.2.21 If G is a gmph with independence nvmber 2 hauing n 2 5 vertices

and rn non-edges, then Z(G) is a dusty subset of the interval [-$,O].

Proof It remaios to show that Z(G) is real and containeci in [- 2, O] . We will do - this by proving that the independence roots of each power Gk ali belong to [-E, O].

First, we prove:

Lemma 3.2.22 The independence roots, Rk, of Gk sutisfy:

Proof of Lemma 3.2.22 The expression for Ri follows directly from equation (3.7)

and the quadratic formula. For each k > 1, we know that we c m find Rk by solving

the equation 2 ma: + nx = Rk-l,

for x, which produces the above expression for Rk.

Next, we can show:

Theorem 3.2.23 For each k 2 1, we have -2 5 RI. 5 0.

Proof of Theorem 3.2.23 By induction on k. For k = 1, we note thst the discrimi-

nant n2 - 4m of (3.7) is 2 O. by applying Twan's theorernt to G. So RI is real. Using

Lemma 3.2.22 and the simple observation that 4m 2 0, we have:

Now suppose the result is true up to before k > 1. Then n2 +4??1&-~ 2 n2 - 4 m - 1 rn =

n2 - 472 = n(n - 4) 2 O. Thus, from Lemma 3.2.22 we have that RI. is real and

and so the result holds for k as well, compieting the proof.

Hence, as the roots at each step are all real and contained in [-2, O], the limiting

set, 1(G) , will also be real and contained in [-+, O] . This completes the proof of

Theorem 3-2-21. n2

t~uran's theorem (cf. (591, p.33) States that for a graph having no triangles, rn 5 - 4 ' where n

and m are its nurnbers of vertices and edges, respectively.

The graph ttj K3 has independence number = 2, n = 5 vertices, and m =

6 non-edges. Then f&) = 6x2 + 52 and, by Theorem 3.2.21, Z(G) is a totdy

disconnected subset of [- %, O], as shown in Figure 3.9.

t1

- 0.5

:I -6.8 4.6 4 . 4 -oT

- 4 . 5

- -1

Figure 3.9: The independence attractor, l ( K z ttj K3) = J(6x2 + 52 ) .

3.2.4 Beyond Independence Number 2

If G is a nonempty graph, and -1 is a mot of i&) = fc(x) + 1 of multiplicity

q(G) > 1, then Theorem 3.2.14 implies that 1(G) is partitioned by Uk>iRoots - (ici)

aad its accumulation points, J( jc) While this implies that Z(G) is disconnected,

it is perhaps more interesting to know whether i ts collection of accumulation points,

J ( fc), is connected.

Perhaps, then, we should reformulate our question. Equation (3.6) t e k us that

the 'new' roots at each step will converge to J(fc), even for v(G) = 1. in fact, the

equation is also valid when i)(G) = 0, since, in that case, Roots icr f7 Roots icl-t = 0.

To see this, suppose the intersection were non-empty. Then, for some z , f ~ ~ ( z ) =

~ ( ~ - ' ) ( r ) ) = ~ ( k - l ) ( t ) = - 1, which says that fc(- 1) = f ~ ( " - ~ ) ( z ) = -1, that is, fc(fc

- 1, contradicting the fact that r](G) = O. We are led to make the following definition.

Definition 3.2.24 For a nonempty graph G , the independence fracta1 T(G) of G is

the set

Z(G) = lim (Roots (ick) \ Roots (ick-i)) = lirn Roots k4aY k-xl

where (as before) q(G) is the muitzplicity of -1 as a mot of ic, and fc = ic - 1.

Thus, T(G) is always J ( fc), whereas Z(G) consists of J( fc) and possibly more,

depending on whether q(G) > 1. And we ask instead:

Question 3.2.25 When is f ( ~ ) connected?

(The name, independence fractal, is not entirely accurate, since not every Julia set

is a Eractal. Recall, for instance, that T ( K ~ - e) = J(x2 + 41) = [-Il O]. A &ta1

(cf. [6]) is generdy considered to be a compact subset of (C,, oo), whose Hausdorfi

dimension (cf. [7]) strictly exceeds its topological dimension.)

If G is a graph with independence number 4 = 3, then

where n, m and t are the numbers of vertices, edges and triangles in C, respectively.

With a Little effort, we can find a conjugate polynomial g&) with no 2- term:

where

m and @(z) = fix + -

3,/T

Since there are now two parameters n and b, the Mandelbrot set (those values of

a and b for which J(x3 + ax + b) is connecteci) is a subset of @ x @, and is much more

complicated than the Mandelbrot set for quadratics. The first extensive study of this

set was carried out jointly by Branner and Hubbard in [16] and [17]. For quartics,

quintics and beyond, very Little is known about the Mandelbrot sets.

The graph G = Cs has independence number 3, and fG(x) = 2x3 + 9x2 + 62. Its

independence attractor is shown in Figure 3.10, and appears to be disconnected. On

the O ther hanci, the graph H whose complement consis ts of t hree triangles intersecting

on a cut vertex, appears to have a connected indpendence fiactal, shown in Figure

3.11.

Independence fractals of graphs of independence number 3 are likely sufficiently

intricate to occupy a thesis in their own right, and we shall not endeavour to investi-

gate them here. Instead, we end the chapter with an analysis of two infinite families

of graphs of arbitrarily high independence numbers.

The following result from iteration theory will be useful. Critical points of a

polynornial play a key role in the connectivity of its Julia set:

Theorem 3.2.26 (cf. [?]) Let f be a polynomial of degree at feast tuto.

a Its Julia set J ( f ) is connected i f and only if the fonuard orbit of each of its

critical points is bounded in (@, 1 1).

a Its Julia set J ( f ) is totnlly disconnected i f (but not only if) the f0TWa~d orbit of

each of i ts critical points is unbounded in (C,[ - 1).

Figure 3.10: The independence attractor, Z(Cs) = f(c6) = J ( k 3 + 9x2 + 6x).

Figure 3.11: The independence attractor, Z ( H ) = T ( H ) = 5(3x3 + 9z2 + 7 4 , where H is the complement of three K3k intersecting on a cut vertex.

The families aKb and Ka, a,. . . , a \

f fl

b

Denote by aKb be the graph Kg kd Kg tfl - tri Kb, the disjoint union of b copies of f

a

the complete graph Ka. The independence attractors of 3K2 and 4K2 are shown in

Figures 3.12 and 3.13, respectively.

We have already described the independence attractors of empty graphs and corn-

plete graphs, and so we may assume that both a and b are greater than 1. Now

Figure 3.12: The independence attractor, 1(3K2).

- a& = K,[Kb], and since fK(x) = (1 + x)" - 1 and fKb(x) = bx, we have

and

fLb(x) = ab(1 + b ~ ) ~ - ' ,

whose only critical point is r = - l /b . By Theorem 3.2.26, then, S(G) will either

be comected or totdy discomected, depending on whether the forward orbit of

t = - l l b is bounded or unbounded, respectively, in (C, 1 1). NOW, faKb(-116) = Oa - 1 = -1.

Case 1: 6 = 2, aeven. Then fdb(z) = f62(x) = (1+2z)"-1. Now fd2(-llb) = -1,

faK2(-1) = (1 - 2)a - 1 = 0, and fd2(0) = O. Hence, the forward orbit of - l /b

converges to 0, and is therefore bounded in (C, 1 1). Thus, f ( a ~ ~ ) is connected.

Case 2: b 2 3, a even. Then f d b ( - t / b ) = -1, and fdb(-1) = (1 - b)" - 1 2 2a - 1 > 1. And z > 1 fa&) > ( 1 +2z)' - 1 = 22 > z + 1. Hence, the forward

orbit of -116 is unbounded in (C, 1 I), and f ( a ~ b ) is totaily disconnecteci.

Case 3: b 2 2 , a odd. Then Then faKb( - l l b ) = -1, and faKb(-1) = (1 - b)" - 1 5 ( 1 -2 )3 - 1 = -2 < -1. Andz < -1 * f(z) < (14-22)' - I = 2 z = r + r < z - 1.

Figure 3.13: The independence attractor, Z(4K2).

Hence, the forward orbit of - l / b is unbounded in (C, 1 I), and ~ ( o K ~ ) is totally

disconnected.

In al1 cases, since b > 1, - 1 is not a fixeci point of faKb, and thus Z(G) = Z(G). We have now proved:

Theorem 3.2.27 The independence attractor of aKb is connected i j b = 2 and a is

euen; totally disconnected othenuise.

As we did for graphs with independence number 2, we can find a region h i d e

which Z(aKb) lies. It lies in the disk

First, we prove:

Lemma 3.2.28 For eaeh k 2 1 , each independence mot, F',, of the graph ( u K ~ ) ~

satisfies:

where wu = Fk-i for some independence mot Fk-i of

Proof The expression for FI cornes kom the fact that ioKb(x) = (1 + bx)=. For each - k > 1, we can find the points Fk by solving the equations

for x, thereby producing the above expression for Fk.

Frorn this, we can prove:

Theorem 3.2.29 For each k 2 1, 1 Fk + b / 5 6.

Proof By induction on k. It is true for k = 1, since F1 = -i irnplies 1 FI + f 1 =

O < f. Now suppose the result is true for a number k - 1 (k > 1). Then. by Lemma

3.2.28, we have:

= - b '

and so the result holds for k as well, completing the proof.

Findy, since roots at each step are at most a distance of from -#, the same

must be true of the limiting set.

Theorem 3.2.30 The independence attzactor of the graph aKs lies in the disk

Further, the bound is best possible. To see this, let z = i (w - 1) be any root u 1 of unity shifted 1 unit to the left, and scaled by i. Then lz + = I i w l = and

foKb(z) = (1 + bz)= - 1 = ( w ) ~ - 1 = 1 - 1 = O. And O E J(fC) for - any nonempty

graph G, since fc (O) = 0, while /&(O) # O, so that O is eit her a repulsive or rationally

indifferent fixed point of fc, thereby lying on J ( fc) Thus, since f&&) = O E J ( jc),

we must have z E J( fc).

Our next lamily is complete multipartite graphs Ka, a , . . . , a = m. Then

Again, we assume that both a and b are greater than 1. Since then -1 is not a fixed

point of fKbLm, we have s (K~[%]) = f ( ~ b [ l ( , ] ) = J (b(1 + r)" - 6).

The independence attractors of the graphs K3,3 = ~ ~ [ m and K4,4 = K ~ [ K ~ ] are

shown in Figures 3.14 and 3.15, respectively. They appear to have the same 'shape'

as the attractors of their cowiterparts, %[K*] and %[K~] (Figures 3.12 and 3.13),

respect ively.

We can explain this phenornenon. Set 4(z ) = bx. Then # is a Mobiiis map, and

Thus, f ~ ~ m and fKb[Kol are analyticdy conjugate (cf. Theorem 3.2.4), and

Figure 3.14: The independence attractor, Z(K3,3).

This observation, together with our analysis of z ( ~ [ K & above, enables us to con-

clude:

Theorem 3.2.31 The independence uttractor of Ka, a , . . , ,a is connected 2f b = 2 - b

and a is even; totally disconnected othenvise. The attractor lies in the disk lz + 11 5 1, and Mis boundzng disk is best possible.

The situation in general is this:

Theorem 3.2.32 For a gmph G ,

where # is the Mobzus rnap x ++ m. Hence,

Figure 3.15: The independence attractor, Z(K4J.

We can also deduce the following.

Theorem 3.2.33 Every nonempty graph G with independence number at least tuo

is an znduced subgmph of a graph H with the same zndependence number, whose

zndependence froctal is disconnected.

Proof Since fG(x) has degree at least 2, an argument similar to that of Lemma 3.2.13 - shows that there exists a r d number R > 1 such that Ir1 > R 1 f&)I > 21~1, which implies that the forward orbit of z is unbounded in (C, 1 1).

Now, not every critical point of fc is a root of 1,. Indeed, for a root r of both f&

and fc, its multiplicity as a root of fc is one greater than its multiplicity as a root

of f&. But deg fc = deg f& + 1, and so, if every critical point of jc were a root of fc,

then in fact fc must have only one critical point c, and jG(x) = a(s + =)p. But we

know that xlfc(x), and so c = O and fc(x) = id. But this could only be the case if

= 1, which it is not.

Let c then be a critical point of fc for which fc(c) = w # 0, and choose a positive

integer n large enough that In - wl > R. For the graph G[K,], we have fciKnl(z) =

fC(m), a critical point of which is c/n. But then fc(rc,l(c/n) = fc(c) = w, and

1 fifKn,(w)I = ( fzk(nw)( -r cm as k -r m. Hence, by Theorem 3.2.26, the graph

C[K,], which has independence number P, and of which G is an induced subgraph,

has a disconnected independence fractal. 0

The argument we used, together with the fact (Theorem 3.2.32) that G[Kn] and

Kn [G] have analytically conjugate independence kactals, enables us to deduce the

following as well, with which we conclude the chapter.

Theorem 3.2.34 If G is a nonernpty graph vith independence numûer at least 2, then

for al1 suficzently large n, the join of n copies of G has a disconnected independence

f~actal.

It would seem, then, that graph connectedness and independence fractd connect-

edness are unrelated.

Chapter 4

Open Problems and Future

Directions

There are many avenues that one could explore.

4.1 Chromatic Roots

Severai conjectures and open problerns came up in our work on roots of chromatic

polynomials in Chapter 2. While we were interested mainly in large subdivisions of a

graph, comput ational explorations wit h real roots of s m d subdivisions suggests the

following.

Conjecture 4.1.1 If euch edge of G is subdivided ut least once, then no real chro-

matzc root of the resulting g ~ a p h G' is greuter than 2.

Here is a partial result in this regard:

Theorem 4.1.2 If we subdivide euch edge of C into a path of even Iength, then no

reul chromatic root of the resulting p p h is 2 o r more.

Proof For convenience, we will denote the chromatic polynomid *(H, z) of a graph - H by the symbol H itself. We argue by induction on the number m of edges in

a graph. For m = 1, the result is clear. Now let m 2 2 and suppose the result

holds for a l l graphs of size at most m - 1. Let G be a graph with rn edges e l , . . . , em.

Subdividing into a path of even length Ii (i = 1, . . . , m), we obtain a graph G;ll"-vem l,....lm ?

whose chromatic polynomial, by equation (2.25), is given by

The number 2 here is best possible, for consider the graph 81,1,i. Among itç even

subdivisions (in the sense of Theorem 4.1.2) are the graphs €3usr,u (1 2 l), whose

chromatic polynomials axe given (cf. equation (2.16) by

With this expression, we can verifjr that , for any given E > 0, the graph 021,z,z

will have a real chromatic root between 2 - E and 2 for 1 sufnciently large.

4.1.1 Bounding Chromatic Roots in terms of Corank

As in [29], let p(G) = max{lt - II: n(G, z ) = O), and, for each positive integer k 2 1,

pk = max{p(G) : graphs G of corad k) . (4.1)

Certainly, po = pl = 1. And pk+l > p k , for if G has corank k, then the disjoint

union of G and a cycle C, is a graph of corank k + 1, whose chromatic roots are

those of G and the cycle. Further, we may restrict our attention to 2-connected

graphs; indeed, if we separate G into its 2-comected components and then glue these

components back together dong a common edge, the resulting g a p h G' has the same

corank as G, and, by the complete intersection theorem mentioned in Chapter 2, has

the same (distinct) chromatic roots as Gr.

Theorem 2.0.2 tells us that pk 5 k for k 1 1, while it foilows from [29] that

h 2 P(k+l;2) = [1 + ~ ( l ) ] k/ logk, where

Whether pk grows linearly or sublinearly is simply not known.

The only 2-connected graphs of corank 2 are 3-ary theta graphs, and so it follows

korn Theorem 2.1.5 that p2 = p ( 2 , 2 , 2 ) 2: 1.5247. Could the same kind of result be

tnie for al1 k? Not quite; in fact, it fails for k = 3. Indeed, while p ( 2 , 2 , 2 ! 2 ) = 1.9636

(cf. Table 2. l), we have p(K4) = 2, since z = 3 is a chromatic root of K4. Nevertheless,

we have found no other counterexample, and so we pose the following.

Conjecture 4.1.3 For al1 k 1 4, pk = p[>t+l;g

Since p(>t+l;l) grows asymptotically as & (cf. [29]), the tmth of this conjecture

would imply that indeed pk exhibits sublinear growth.

4.2 Independence Attractors

The relationships between a graph and its independence fracta1 remains a tantaking

question. Even the restncted question of when an independence fracta1 is connected

seems elusive. Certainly, it does not depend on the connectivity of the gaph. We

have seen, for instance, that 4K2, a disconnecteci graph, has a comected independence

kactd, while Theorem 3.2.34 parantees the existence of many connected graphs with

dkcomected independence fiactals. We were able to provide a complete answer for

gaphs with independence number 2 in Section 3.2.3; it may be possible to complete

the p = 3 picture as well.

Just how much about a graph can its independence attractor tell us? Theorem

3.2.32 tells us that G[K,] and K,[G] have analytically conjugate independence frac-

tals. Further, since (cf. Section 3.2.1) for any polynomial f , its Julia set is equal

to that of fok for any k 2 1, it follows that Gk and G have identicai independence

fractals. These obversations give a partial answer to the following

Question 4.2.1 When do two graphs G and H houe analytically conjugate indepen-

dence fractals?

4.2.1 Independence Attractors of Complexes

The independent sets of vertices in a graph lorm a cornplex. An (abstract, simplicial)

complex on a set X is a collection C of subsets of X that is closed under containment,

i.e., if A E C and B C A, then B E C. The sets in C are the faces of the complex.

and faces of cardinality one (the elements of X) are the vertices of C. Denoting by f i

the number of faces in C of cardinality i, define the f -polynomial of C as

For a gaph G, the collection C of independent sets of vertices in G is indeed a

cornplex on V(G), as any subset of an independent set is again an independent set;

and

f c ( 4 = j ~ ( x ) = i ~ ( x ) - 1. For two complexes C and V on sets X and Y, respectively, we can dehe their com-

position C[V] as the complex on X x Y whoçe faces are all sets C x D, where C E C and D E V. The argument we gave for Theorem 3.2 also proves the following:

Thus, by leaving out the constant term fo = 1, the polynomials are closed under

composition. Define the uttractor of a complex C as

Frac (C) = lim Roots ( fci) . k+da

This tirne, the attractor is the inverse orbit of O, instead of - 1. But O is a fixed point

of fc, and is not a root of fi, which irnplies that O E J ( fc), and so

iliustrating a second advantage to leaving out the constant term: the attractor never

contains anything but the Julia set. This does corne with a small price: in terms

of f - polynomials, Theorems 3.0.13 and 3.0.14 now have the slightly more awkward

forrn

f c ( 4 = je-v(4 + x . ( f c - [ V I (4 + 1) and

fcm(4 = (fc(z) + 1) (f&) + 1) - 1, where, for v a vertex of C, C - v consists of all faces in C that do not contain u, and

C - [u] those faces in C containing neither v nor any vertex which did not lie in a face

(in C) with v .

Probably the simplest example of a complex is an nsimplex, C,,, which consists

of an n element set, together with all of its subsets. Since, for each i, fi is then the

binomial coefficient C (n, i), the f -polynomial has the form fc(x) = (1 + x)" - 1.

And then, for each k 1 1, fzk(z) = (1 + z ) ~ - 1, whose roots are the kn-th roots of

unity, shifted to the left one unit. As k -t oo, the roots become dense on the circle

Ir + 11 = 1, and hence

Theorem 4.2.2 For an n-simplez CR,

Frac (C,) = {Z : lz + il = 1).

Now, if we take an nsimplex Cn and remove one of its faces F of cardinality

m < n (dong with dl faces in Cn containing F), we obtain a complex dm), say,

whose f -polynomial can be seen as foUows. For each i l the number of faces in C, of

cardinality i which contain F is the binomial coefficient C (n - m, i - m). Hence,

If, for instance, C4 is the 4- sirnplex on {a, b, c, d) , and cY) the complex obtained by

rernoving face {a, 6 ) (and those containing it), then cY) is the following cornplex:

And fqi(x) = ( 1 + ~ ) ~ - X ~ ( ~ + X ) ~ - 1 = 2 x 3 + 5 x 2 + k . Its attractor, Rac C, ( (*))

is shown in Figure 4.1. In addition, plots of the attractoa Frac (cY)) and Rac (cf)) are found in Figures 4.2 and 4.3, respectively.

Let us conclude with a proof of the following.

Theorem 4.2.3 The complexes &, &) and d2) al1 have connected attractors.

Proof Theorem 4.2.2 tells us that Frac (G) is connected. Since &) is nothing but

Çr-i, then its attractor is also connected. Now consider k2), where n >_ 3. Then

Figure 4.1: The attractor, Rac (c?)) .

and (by a simple calculation)

whose critical points are -1 and -. If n = 3, then

whose Julia set, as we saw in Section 3.2.3, is connected.

We may assume, then, that n 2 4. Now fkll (-1) = -1, and (the reader can

verify t his)

Figure 4.2: The attractor, Frac ( C, (3)) .

And (x + 11 < E < f implies

which is c E " - ~ - 2 5 2 - 2 c E - a 2 = a€. Hence, the forward orbit of a wiU

converge to -1, and is therefore bounded in (C, 1 1). Thus, since d critical points

of f& have baunded forward orbits, its Julia set, Frac (ci2)), is connected. O

Figure 4.3: The attractor, Frac C, . ( (4))

Figures 4.2 and 4.3 show that cY) is connected, while c?) is not. Perhaps it iç

true in generd that for some k (possibly dependeing on n), &') is disconnected for

each j 1 k. Computationd explorations do suggest that &' is always disconnected.

Appendix A

A Maple Procedure for

Independence At t ract ors

Below is the procedure the author used to constnict his plots of independence at-

tractors. It takes as input a polynomial f E Z[x] (the f - polynomial of a graph or

cornpleu), a starting point z (where z = -1 for graphs. and r = O for complexes) and

a 'mesh number' N, to be defined momentarily. It then constructs (the begimings

of) the backwaxd orbit O-(2) as follows.

First, an imaginary grid is placed over the complex plane: its squares have side

1/N. Whenever an inverse image X is computed, the grid squares in which the points

of X reside are determined. Those points which land in a square previously visited

are then discardeci, so that their inverse images are not taken.

This is a standard tri& (cf. [a]) for computing Julia sets in practice. It dows one

to go deeper into the backward orbit, whose sets are actually growing wponentially

in size. The result is a more complete, more uniform plot.

The algorithm terminates when all points at given a aven step are discarded.

Thst this happens is an immediate consequence of the fact (cf. Section 3.2.1) that

Julia sets of polynomials are bounded in (@, 1 - 1).

Attract := proc (f , z,N)

local A,u,r,xx,yy,OldRts,NewRts,pts,i,j,n,symb:

n := degee(f):

A := table():

pts := MILL:

OldRts : = [op ({f solve (f , x , cornplex) ) minus (2)) 1 :

pts := pts, [Re(z) ,Im(z)] ,op(rnap([Re,Im] ,OldRts)) :

while nops(0ldRts) > O do

NewRt s : = NULL :

for i from 1 to nops(0ldRts) do

v : = [f solve (f -0ldRts [il , x , cornplex) 1 : for j from 1 to n do

r := wCjJ:

xx := ceil(N*(Re(r))):

yy := ceil(N*(Im(r)) :

if A [ x x , y y ] o l then

A[xx,yy] := 1:

NewRts := NewRts,r:

pts := pts, [Re(r) , Im(r)l :

fi:

od:

od:

OldRt s : = [NeuRt s] :

od:

pts := [pts] :

symb := cross:

for i from 1 to min(nops(pts),lO) do

if pts Ci] [2] 0 O then

symb := point:

i : = min(nops (pts) ,20) :

fi:

od:

if symb=point then

print('Poirits calculated. Plotting ...O : else

print('Attractor is Real. Plotting. . . ' ) :

fi:

plot(pts,style=point,symbol=symb,colour=black,scaling=constrained);

end ;

The latter part of the algorithm simply looks at the first 20 points in the List

obtained, to "determine" whether the attractor is real. An appropriate plotting

symbol is then chosen.

The plot in figure 3.7, for instance, namely l ( P 3 ) = J(x2 + 3 1 ) was obtained with

the command > At t rac t (xS+3*x, -1, LOO) ;

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