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

  • Wayne Barrett
  • Shaun Fallat
  • H. Tracy Hall
  • Leslie Hogben
  • Jephian C.-H. Lin
  • Bryan L. Shader
Keywords: Inverse Eigenvalue Problem, Strong Arnold Property, Strong Spectral Property, Strong Multiplicity Property, Colin de Verdière type parameter, Maximum multiplicity


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

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