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

2015/11/20 by Wayne Barrett, Shaun Fallat, Barrett, Wayne +9 · 3 citations
Computer Science · Engineering · Mathematics · #05C50 #15A18 #15A29 #15B57 #58C15 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1511.06705

openalex publication_date 2015/11/20 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

For a given graph G and an associated class of real symmetric matrices whose\noff-diagonal entries are governed by the adjacencies in G, the collection of\nall possible spectra for such matrices is considered. Building on the\npioneering work of Colin de Verdiere in connection with the Strong Arnold\nProperty, two extensions are devised that target a better understanding of all\npossible spectra and their associated multiplicities. These new properties are\nreferred to as the Strong Spectral Property and the Strong Multiplicity\nProperty. Finally, these ideas are applied to the minimum number of distinct\neigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at\nleast the number of vertices of G less one are characterized.\n

Cited by

• The Strong Singular Value Property for Matrices
• Inverse eigenvalue problem for discrete Schrödinger operators of a graph
• The minimum number of distinct eigenvalues of a threshold graph is at most 4

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