Reflexivity in some classes of multicyclic treelike graphs

1. Reflexivity in some classes of multicyclic treelike graphs Bojana Mihailović, Zoran Radosavljević, Marija Rašajski Faculty of Electrical Engineering, University of Belgrade, Serbia

2. 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

3. Introduction • Being reflexive is a hereditaryproperty • Presentation of all reflexive graphs inside given set: via maximal graphs or via minimal forbidden graphs • Smith graphs

4. 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

5. 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

6. 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

7. 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 .

8. 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).

9. First results

10. 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.

11. 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

12. 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

13. 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?

14. 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 )

15. New results/bundle

16. 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

17. 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

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

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

20. New results/bundle • 1. case • 2. case

21. New results/bundle • 1. case Maximal number of cycles is 74. • 2. case Maximal number of cycles is 22.

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.