Reflexivity in some classes of multicyclic treelike graphs

Bojana Mihailović, Zoran Radosavljević, Marija Rašajski

Faculty of Electrical Engineering, University of Belgrade, Serbia

Introduction

Graph = simple graph (finite, nonoriented, without loops and/or multiple edges) + connected graph

Spectrum = spectrum of (0,1) adjacency matrix (the spectrum of a disconnected graph is the union of the spectra of its components)

A graph is treelike or cactus if any pair of its cycles has at most one common vertex

A graph is reflexive if its second largest eigenvalue does not exceed 2

Introduction Being reflexive is a hereditary property

Presentation of all reflexive graphs inside given set: via maximal graphs or via minimal forbidden

graphs

Smith graphs

1

3

2

n n -1

C n W n

1 2 n

Instruments

Interlacing theorem Let A be a symmetric matrix with eigenvalues and B one of its principal submatrices with eigenvalues Then the

inequalities hold. Schwenk’s formulae newGRAPH

1,..., n 1,..., .m

( 1,..., )n m i i i i m

Instruments

RS theorem Let G be a graph with a cut-vertex u. 1) 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

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

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

u

1G2G 3G nG

2 ( ) 2.G

2 ( ) 2.G

2 ( ) 2.G

G

First results Class of bicyclic graphs with a bridge between

two cycles of arbitrary length Additionally loaded vertices which belongs to

the bridge – 36 maximal graphs Also additionally loaded other vertices – 66

maximal graphs

First results Splitting

If we form a tree T by identifying vertices x and y of two trees and , respectively, we may say that the

tree Tcan be split at its vertex u into and .

x y uT 1 T 2 T 1 T 2

( )x = y = u1T 2T

1T 2T

First results PouringIf we split a tree T at all its vertices u, in all possible ways, and in each case attach the parts at splitting vertices x and y to some vertices u and v of a graph G (i.e. identify x with u andy with v), we say that in the obtained family of graphs the tree T is pouring between the vertices u and v (including attaching of the intact tree T, at each vertex, to u or v).

GT 1 T 2u v

First results

1S

2S

Multicyclic treelike reflexive graphsUnder 2 conditions: cut vertex theorem can not be applied cycles do not form a bundletreelike reflexive graph has at most 5 cycles.

Multicyclic treelike reflexive graphsUnder previous 2 conditions all maximal reflexive cacti with four cycles are

determined four characteristic classes of tricyclic reflexive graphs

are defined class is completely described via maximal graphs

4l

4K

4K3K2K1K

New results/current investigations

classes and are completely described some new interrelations between these

classes and certain classes of bicyclic and unicyclic graphs are established

some results are generalized

1K 3K

New results/bundle cut-vertex theorem can

not be applied, but cycles do form a bundle

after removing vertex v one of the components is a supergraph and all others subgraphs of some Smith tree

If G is reflexive, what is the maximal number of cycles in it?

1C

2C

nC

1T

2TmT

v

G

New results/bundle K = the component of the graph G-v which is a

supergraph of some Smith tree K = minimal component e.g. for every its vertex x,

whose degree in the graph G is 1, condition

holds 2 cases:

1. K is a subgraph of the cycle C (C is additionally loaded with some new edges)

2. K is a subgraph of the tree T K must contain one of the F - trees (minimal

forbidden trees for )

1 1( ) 2 ( ) (1)K x K

2 2

New results/bundle

1F 2F 3F 4F 5F

6F7F 8F 9F

xx x x x

x x x x

New results/bundle

1. caseBlack vertices are the vertices of K adjacent to vertex v. both black vertices belong to the same F-tree

one black vertex belong to F-tree, and the other doesn’t

i) any vertex of F-tree different from x may be black vertex

ii) extended with additional path at vertex x

iK = F

K = F

i 4 8 7 3 2 9

path length

1 1 1,2 1,2,3

arb. arb.

New results/bundle

2. caseIt is sufficient to discuss the case when T-v has one

component K.Black vertex d is a vertex of K adjacent to v. d belongs to F-tree

i) any vertex of F-tree different from x may be black vertex

ii) K=F

Both cases 2(2) 0 ( ) 2GP G

New results/bundle 1. caseC – cycle which contains K; v – cut vertex; x,y – black

vertices 1 1

1 2 1 2 1 1

1 2 1 3 1 1

1

(2) 2 (2) ... ( (2) (2)) ...

2 (2)(( 1) ... ( 1)... ... ... ( 1))

2 (2) ... 2 (2)( ... ... ... ... )

... (2 (2) (2)m

G C v k C v x C v y k

C v k k k k

C C k C v k k k

k C v C v x C v

P P n n P P n n

P n n n n n n n n n

P n n P n n n n n n n

n n P P P

(2) 2 (2) 2 (2))my C v C CkP P

(2) 0 (2) 2 (2) 0G C C vP P kP

New results/bundle 2. caseT-v=K; v – cut vertex

1 1 2

1 2 1 1 1

2 1 3 1 1

1

(2) 2 ... (2) 2 (2)(( 1) ...

( 1)... ... ... ( 1)) (2) ...

2 (2)( ... ... ... ... )

... (2(1 ) (2) (2))

G k K K k

k k k K d k

K k k k

k K K d

P n n P P n n n

n n n n n n P n n

P n n n n n n n

n n k P P

(2) 0 2(1 ) (2) (2) 0G K K dP k P P

1F 2F 3F 4F 5F 6F 7F 8F 9F

4 10 12 13 13 13 20 34 74

New results/bundle 1. case

2. case

1F 2F 3F 4F 5F 6F 7F 8F 9F

2 4 4 4 4 4 7 11 22

New results/bundle 1. case

Maximal number of cycles is 74.

2. case

Maximal number of cycles is 22.

References D. Cvetković, L. Kraus, S. Simić: Discussing graph theory

with a computer, Implementation of algorithms. Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz. No 716 - No 734 (1981), 100-104.

B. Mihailović, Z. Radosavljević: On a class of tricyclic reflexive cactuses. Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 16 (2005), 55-63.

M. Petrović, Z. Radosavljević: Spectrally constrained graphs. Fac. of Science, Kragujevac, Serbia, 2001.

Z. Radosavljević, B. Mihailović, M. Rašajski: Decomposition of Smith graphs in maximal reflexive cacti, Discrete Math., Vol. 308 (2008), 355-366.

Z. Radosavljević, B. Mihailović, M. Rašajski: On bicyclic reflexive graphs, Discrete Math., Vol. 308 (2008), 715-725.