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

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

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 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 T can be split at its vertex u into and.

First results Pouring If 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 and y 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).

First results

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

Multicyclic treelike reflexive graphs Under 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

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

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?

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 )

New results/bundle

1. case Black 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 i path length 111,21,2,3arb.

New results/bundle 2. case It 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

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

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

New results/bundle 1. case 2. case

New results/bundle 1. case Maximal number of cycles is case Maximal number of cycles is 22.

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