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

Date
2017-01-01
Authors
Barrett, Wayne
Fallat, Shaun
Hall, H. Tracy
Hogben, Leslie
Lin, Jephian
Shader, Bryan
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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.

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This article is published as W. Barrett, S. Fallat, H. T. Hall, L. Hogben, J. C.-H. Lin, B.L. Shader. Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph. Electronic Journal of Combinatorics 24, no. 2 (2017): P2.40.

https://www.emis.de/journals/EJC/ojs/index.php/eljc/article/view/v24i2p40/0

Keywords
Inverse Eigenvalue Problem, Strong Arnold Property, Strong Spectral Property, Strong Multiplicity Property, Colin de Verdière type parameter, Maximum multiplicity
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