ON UNICYCLIC REFLEXIVE GRAPHS

• The spectrum of a simple graph (non-oriented, without loops and multiple edges) is the spectrum of its adjacency matrix, along with the usual assumption

• The Interlacing theorem: Let be the eigenvalues of a

graph G and eigenvalues of its induced subgraph H. Then the inequalities

, , hold.• Reflexive graphs are graphs having

1 2( , ,..., )n 0,1

1 2 ... .n

1 2 ... n

1 2 ... k

n k i i i 1,2,...,i k

2 2.

• Graph G is a maximal reflexive graph inside a given class of graphs C if G is reflexive and any extension G+v that belongs to C has

• Theorem (Smith): For a simple graph G (resp. )

if and only if each component of G is an induced subgraph (resp. proper induced subgraph) of one of the graphs of Fig. 1, all of which have

2 2.

1 2G 1 2G

1 2.G

Figure 1.Connected graphs that have their largest eigenvalue (the index) equal to 2 are known as Smith graphs

1

1

3

2

n n -1

C n W n

1 2 n

• Theorem (Schwenk): Given a graph G, let ( ) denote the

set of all cycles containing a vertex and an edge of G, respectively. Then

(i)

(ii)

where Adj(v) denotes the set of neighbors of v, while G – V(C) is the graph obtained from G by removing the vertices belonging to the cycle C.

C v C uvv

uv

2 ,G G v G v u G V Cu Adj v C C v

P P P P

2 ,G G uv G v u G V CC C uv

P P P P

• Corollary 1. Let G be a graph obtained by joining a vertex of a

graph to a vertex of a graph by an edge. Let ( ) be the subgraph of ( ) obtained by deleting the vertex ( ) from (resp. ). Then

• Corollary 2. Let G be a graph with a pendant edge , being of degree 1. Then where ( ) is the graph obtained from G (resp. ) by deleting the vertex (resp. )

1v

1G 2v 2G

1G 2G

1G 2G1v 2v 1G 2G

1 2 1 2' ' .G G G G GP P P P P

1 2v v 1v

1 2

,G G GP P P

1G 2G

1v 2v1G

Theorem RS

Let G be a graph with a cut vertex u.i. If at least two components of G-u are supergraphs

of Smith graphs, and if at least one of them is a proper supergraph, then

ii. If at least two components of G-u are Smith graphs, and the rest are subgraphs of Smith graphs, then

iii. If at most one component of G-u is a Smith graph, and the rest are proper subgraphs of Smith graphs, then

2 2.G

2 2.G

2 2.G

Maximum number of loaded vertices of the cycle

in unicyclicreflexive graph

Theorem 1.

The cycle of unicyclic reflexive graph of length greater than 8 cannot have more than 7 loaded vertices.

Theorem 2.

The cycle of unicyclic reflexive graph of length greater than 10 cannot have more than 6 loaded vertices.

The length of the cycle with six loaded vertices

Theorem 3. Maximal length of the cycle of unicyclic reflexive graph with 6

loaded vertices is l = 12.

The length of the cycle with five loaded vertices

Theorem 4.

1. Maximal length of the cycle of unicyclic reflexive graph with 5 loaded vertices, if these vertices are not consecutive, is l = 14.

2. Maximal length of the cycle of unicyclic reflexive graph with 5 consecutive loaded vertices is l = 16.

The length of the cycle with four loaded vertices

Theorem 5.

1. Maximal length of the cycle of unicyclic reflexive graph with 4 loaded vertices, if there are no consecutive loaded vertices on the cycle, is l = 16.

2. Maximal length of the cycle of unicyclic reflexive graph with 4 loaded vertices, if there are are two (but not three, and not four) consecutive loaded vertices on the cycle is l = 21.

3. Maximal length of the cycle of unicyclic reflexive graph with 4 loaded vertices, if there are are three (but not four) consecutive loaded vertices on the cycle is l = 38.

4. The length of the cycle of unicyclic reflexive graph with 4 consecutive loaded vertices, has no upper bound.

Let m, p, n, q be the lengths of the paths (a; b), (c; d), (a; d), (b; c), respectivelyThe length of the cycle of graph G is l = m + p + q + n

PG(2) = mpqn-4mpn-4mnq-4pqm- 4pqn+12mn+12mq+12pq+12np+16nq+16mp-32m - 32n - 32p - 32q

m = 1: PG(2) = -3pqn + 8pn + 12nq + 8pq - 20n - 20q - 16p - 32 n

m n q PG(2) λ2≤2 λ2>2 l1. 1 2 2 4p-64 p≤16 p≥17 212. 1 2 3 6p-60 p≤10 p≥11 163. 1 2 4 8p-56 p≤7 p≥8 144. 1 2 5 10p-52 p≤5 p≥6 135. 1 2 6 12p-48 p≤4 p≥5 136. 1 2 7 14p-44 p≤3 p≥4 137. 1 2 8 16p-40 p≤2 p≥3 138. 1 2 9 18p-36 p≤2 p≥3 149. 1 2 10 20p-32 p≤1 p≥2 14

10. 1 2 11 22p-28 p≤1 p≥2 1511. 1 2 12 24p-24 p≤1 p≥2 1612. 1 3 3 5p-44 p≤8 p≥9 1513. 1 3 4 4p-28 p≤7 p≥8 1514. 1 3 5 3p-12 p≤4 p≥5 1315. 1 3 6 2p+4 / p≥1 /16. 1 3 7 p+20 / p≥1 /17. 1 3 8 36 / p≥1 /

m = n = 1: PG(2) = 5pq - 8p - 8q – 52

p≥35: PG(2)= 167q - 332 > 0

p≥11 and q≥3: PG(2)= 47q - 140 > 0

p PG(2) λ2≤2 λ2>2 l

1. 2 2q-68 q≤34 q≥35 38

2. 3 7q-76 q≤10 q≥11 15

3. 4 12q-84 q≤7 q≥8 13

4. 5 17q-92 q≤5 q≥6 12

5. 6 22q-100 q≤4 q≥5 12

6. 7 27q-108 q≤4 q≥5 13

7. 8 32q-116 q≤3 q≥4 13

8. 9 37q-124 q≤3 q≥4 14

9. 10 42q-132 q≤3 q≥4 15

m=n=p=1: PG(2) = -3q - 60 < 0

The length of the cycle with three loaded vertices

Theorem 6. Let G be the unicyclic reflexive graph with exactly three

loaded vertices of the cycle, and let m, n and k be the lengths of the paths between its loaded vertices, p= min (m,n,k).

1. If p≥3 then the maximal length of the cycle is 18. 2. If p=2: 2.1. m=n=2, the length of the cycle is not bounded. 2.2. m=2, n≥3, k≥3, maximal length of the cycle is 23. 3. If p=1: 3.1. m=n=1, or m=1, n=2, the length of the cycle is not bounded. 3.2 m=1, n≥3, k≥3, maximal length of the cycle is 40.

Let m, n, k be the lengths of the paths (a; b), (b; c), (c; a), respectively

The length of the cycle of graph G is l = m + n + k

p = min (m,n,k)

PG(2) = -mnk + 4mn + 4mk + 4nk - 12m - 12n - 12k

p=1: p=3:

p=2: p=4:

m n k l1 3 k≤36 7≤l≤401 4 k≤11 8≤l≤161 5 k≤7 9≤l≤131 6 k≤6 10≤l≤13

m n k l2 3 k≤18 8≤l≤232 4 k≤10 9≤l≤162 5 k≤7 10≤l≤142 6 k≤6 11≤l≤14

m n k l

3 3 k≤12 9≤l≤18

3 4 k≤9 10≤l≤16

3 5 k≤7 11≤l≤15

3 6 k≤6 12≤l≤15

m n k l

4 4 k≤8 12≤l≤16

4 5 k≤7 13≤l≤16

4 6 k≤6 14≤l≤16

p=5:1. m=n=k=5, l=152. m=n=5, k=6, l=163. m=5, n=k=6, l=17

p=6:m=n=k=6, l=18