Campus Units

Mathematics

Document Type

Article

Publication Version

Submitted Manuscript

Publication Date

2017

Journal or Book Title

arXiv

Abstract

The inverse eigenvalue problem of a given graph G is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [8]. In that paper it was shown that if a graph has a matrix with the SSP (or the SMP) then a supergraph has a matrix with the same spectrum (or ordered multiplicity list) augmented with simple eigenvalues if necessary, that is, subgraph monotonicity. In this paper we extend this to a form of minor monotonicity, with restrictions on where the new eigenvalues appear. These ideas are applied to solve the inverse eigenvalue problem for all graphs of order five, and to characterize forbidden minors of graphs having at most one multiple eigenvalue.

Comments

This is a pre-print of the article Barrett, Wayne, Steve Butler, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Jephian C.-H. Lin, Bryan L. Shader, and Michael Young. "The inverse eigenvalue problem of a graph: Multiplicities and minors." arXiv preprint arXiv:1708.00064v1 (2017). Posted with permission.

Copyright Owner

The Authors

Language

en

File Format

application/pdf

Published Version

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