Generalizations of the Strong Arnold Property and the Minimum Number of Distinct Eigenvalues of a Graph

Abstract

For a given graph G and an associated class of real symmetric matrices whose offdiagonal 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 Verdière 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.

Sign patterns of orthogonal matrices and the strong inner product property

“…The terminology SIPP follows that of the strong Arnol'd property which uses similar ideas to obtain results about the maximum nullity of a certain family of symmetric matrices associated with a graph [10]. There have been other generalizations of the strong Arnol'd property in different settings [4].…”

“…U is an open set in R d and for each t ∈ (−1, 1) the function ϕ(·, t) is a diffeomorphism between U and the manifold M (t). Theorem 4.1 below is a specialization of Lemma 2.2 in [10] and is stated with proof in [4].…”

Even maps, the Colin de Verdière number and representations of graphs

“…It is not difficult to see that one can transfer this statement into a formula in the language L. Additionaly, one gets easily an ∃∀-sentence of length polynomial in n. The reason for the presence of the universal quantifier is the definition of SAH, which is a condition on all matrices of certain form. The main ingredient in changing this formula into an existence formula is the following equivalent characterization of SAH by Barrett et al [Bar+17]:…”

On Orthogonal Matrices with Zero Diagonal

“…Much recent research has been devoted to such classes of matrices (see, for instance, the surveys by Fallat and Hogben [11,12]). In particular, in [1,2,10,13,17] the following question is considered: for a given graph G, what is the minimum number of distinct eigenvalues of a matrix in S(G)? This parameter is denoted q(G).…”