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

TL;DR

This paper introduces the Strong Spectral Property and Strong Multiplicity Property as extensions of the Strong Arnold Property, providing new insights into the spectra and eigenvalue multiplicities of matrices associated with graphs, and characterizes graphs with high minimum eigenvalue counts.

Contribution

It proposes two new properties extending the Strong Arnold Property to better understand spectra and eigenvalue multiplicities in graph-associated matrices.

Findings

Characterization of graphs with q(G) at least |V(G)| - 1

Introduction of the Strong Spectral and Multiplicity Properties

Insights into the spectrum of matrices related to graphs

Abstract

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.