Dragoš Cvetković

The largest eigenvalue of a graph: A survey

This article is a survey of results concerning the largest eigenvalue (or index) of a grapn, catcgoiizeu as follows (1) inequalities lor the index, (2) graph with bounded index, (3) ordering graphs by their indices, (4) graph operations and modifications, (5) random graphs, (6) applications.

Variable Neighborhood Search for Extremal Graphs 2. Finding Graphs with Extremal Energy

The recently developed Variable Neighborhood Search (VNS) metaheuristic for combinatorial and global optimization is outlined together with its specialization to the problem of finding extremal graphs with respect to one or more invariants and the corresponding program (AGX). We illustrate the potential of the VNS algorithm on the example of energy E, a graph invariant which (in the case of molecular graphs of conjugated hydrocarbons) corresponds to the total π-electron energy. Novel lower and upper bounds for E are suggested by AGX and several conjectures concerning (molecular) graphs with extremal E values put forward. Moreover, most of the bounds are proved to hold.

A table of connected graphs on six vertices

Abstract This paper contains a table of 112 connected graphs on six vertices. The graphs are ordered lexicographically by their spectral moments in non-increasing order. The pictures of graphs are given to show as much symmetry as possible. Several data such as the spectrum, and its main part, coefficients of the characteristic and of the matching polynomial, numbers of circuits, etc., are given for each graph in the table. Several observations implied by this table are noted.

Graph theory and molecular orbitals. VII. The role of resonance structures

The relations between the simplest variants of MO and VB theory are discussed. It is shown that there is a unique principle causing all the cases of congruity between these two theories‐Kekule structures being related to the permutations contained in the molecular graph [Eqs. (6) and (7)]. The class of benzenoid hydrocarbons where both theories are substantially equivalent is rigorously defined using graph theory. A number of topological regularities for these hydrocarbons are proved. Thus, the Dewar‐Longuet‐Higgins equation, the proof of the Ruedenbergs and Paulings bond orders, the relation between the VB and MO spin and charge density, and Heilbronners formula are obtained. The limits of validity for all these results are strictly determined.

Spectra of unicyclic graphs

Unicyclic graphs are discussed in the context of graph orderings related to eigenvalues. Several theorems involving lexicographical ordering by spectral moments as well as the ordering by the largest eigenvalue are proved. An appendix contains a table of the 89 unicyclic graphs on eight vertices together with their spectra, spectral moments and characteristic polynomials.

Kekulé structures and topology

Abstract The connection between the number of Kekule structures and molecular topology is discussed.

On spectral characterizations and embeddings of graphs

The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G. In this paper all graphs with λ(G) greater than −2 are characterized. Such a graph is a generalized line graph of the form L(T;1,0,…,0), L(T), L(H), where T is a tree and H is unicyclic with an odd cycle, or is one of 573 graphs that arise from the root system E8. If G is regular with λ(G)>−2, then Gis a clique or an odd circuit. These characterizations are used for embedding problems; λR(H) = sup{λ(G)z.sfnc;H in G; G regular}. H is an odd circuit, a path, or a complete graph iff λR(H)> −2. For any other line graph H, λR(H) = −2. A similar result holds for complete multipartite graphs.

Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have been formulated after some experiments with the programming system AutoGraphiX, designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search. The conjectures are related to the maximal value of the irregularity and spectral spread in n-vertex graphs, to a Nordhaus-Gaddum type upper bound for the index, and to the maximal value of the index for graphs with given numbers of vertices and edges. None of the conjectures has been resolved so far. We present partial results and provide some indications that the conjectures are very hard.

Developments in the theory of graph spectra

Some of the main directions and newer results in the theory of graph spectra are reviewed. Some areas discussed are (i) the application of root systems to the theory of graph spectra, (ii) results concerning spectral characterizations of graphs with least eigenvalue-2, (iii) a description of some new graph invariants based on the eigenvectors of the adjacency matrix, along with (iv) the algebraic solution of the Shannon capacity problem, (v) results on spectra of random graphs, and (vi) a review of some graph polynomials related to the characteristic polynomial. Finally, (vii) a discussion of recent results concerning the spectra of infinite graphs is given.

Variable neighborhood search for extremal graphs 3

In the set of bicolored trees with given numbers of black and of white vertices we describe those for which the largest eigenvalue is extremal (maximal or minimal). The results are first obtained by the automated system AutoGraphiX, developed in GERAD (Montreal), and verified afterwards by theoretical means.