Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdiere in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at least the number of vertices of G less one are characterized.Comment: 26 pages; corrected statement of Theorem 3.5 (a

Expected values of parameters associated with the minimum rank of a graph

We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdière-type parameters. Let G(v,p) denote the usual Erdős-Rényi random graph on v vertices with edge probability p. We obtain bounds for the expected value of the random variables mr(G(v,p)), M(G(v,p)), ν(G(v,p)) and ξ(G(v,p)), which yield bounds on the average values of these parameters over all labeled graphs of order v