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Reflexivity in some classes of multicyclic treelike graphs. Bojana Mihailovi ć , Zoran Radosavljevi ć , Marija Ra š ajski Faculty of Electrical Engineering, University of Belgrade, Serbia. Introduction.
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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 hereditaryproperty • 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 componentsofG-uare supergraphsof Smith graphs, and if at least one of them is a proper supergraph, then 2) If at least two components ofG-u are Smith graphs and the rest are subgraphs of Smithgraphs, then 3) If at most one component ofG-u is a Smith graph, and the rest are proper subgraphs ofSmith graphs, then G u
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 Tby 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 vof 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).
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 = minimalcomponente.g. for every its vertex x, whose degree in the graph G is 1, condition holds • 2 cases: • K is a subgraph of the cycle C (C is additionally loaded with some new edges) • 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 • any vertex ofF-tree different fromx may be black vertex • extended with additional path at vertex x
New results/bundle 2. case It issufficient 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 • any vertex ofF-tree different fromx may be black vertex • 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 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.