Open problems on graph eigenvalues studied with AutoGraphiX

Abstract

Since the late forties of the last century, methods of operations research have been extensively used to solve problems in graph theory, and graph theory has been extensively used to model operations research problems and to solve optimization problems on graphs, e.g., shortest paths and network flow problems. More recently, methods of operations research and artificial intelligence have been used to advance graph theory per se, i.e., to find conjectures on graph theory invariants, to refute such conjectures and in some cases to find automated proofs or ideas of proofs. Among other systems, the AutoGraphiX system was developed since 1997 at GERAD (Montreal) by the present authors. Extensive experiments have been conducted which led to 1,700 conjectures, about 800 of which turned out to be easy and could be proved by the system, and about 600 further ones were proved by hand by us or graph theorists from various countries. Moreover, these results led to many generalizations and further papers. In this paper, we study four theoretical problems related to the eigenvalues of (the adjacency matrix of) a connected graph and to which AutoGraphiX was applied. Three of the problems are related to the maximum value of the irregularity, the maximum spectral spread and the upper bound of Nordhaus-Gaddum type on the index, over the class of connected graphs on n vertices. The fourth problem concerns the maximization of the energy (the sum of the absolute values of the eigenvalues) of a connected graph with fixed numbers of vertices and of cycles. We present a brief survey of the papers on or in connection with these problems, and give some new partial results.

Nordhaus–Gaddum and other bounds for the sum of squares of the positive eigenvalues of a graph

“…The authors of [3,5] used AGX to devise the following conjecture. Before the statement of the conjecture, recall that a complete split graph CS n,α on n vertices with independence number α is the complement of the graph composed of a clique K α and n−α isolated vertices.…”

“…which implies equality in(5), i.e., µ √ s − = m, and thenµ = √ s − = √ m.Thus, s − = m and therefore µ 2 = m = s + , which means that G contains exactly one positive eigenvalue, i.e. G is necessarily a complete multipartite graph.…”

On a conjecture about tricyclic graphs with maximal energy

“…Theorem 1.2 Among all unicyclic graphs on n vertices, the cycle C n has maximal energy if n ≤ 7 but n = 4, and n = 9, 10, 11, 13 and 15 ; P 3 4 has maximal energy if n = 4 . For all other values of n , the unicyclic graph with maximal energy is P 6 n .…”