The inverse eigenvalue problem of a graph: Multiplicities and minors
Abstract
The inverse eigenvalue problem of a given graph is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in . 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.
Cite
@article{arxiv.1708.00064,
title = {The inverse eigenvalue problem of a graph: Multiplicities and minors},
author = {Wayne Barrett and Steve Butler and Shaun M. Fallat and H. Tracy Hall and Leslie Hogben and Jephian C. -H. Lin and Bryan L. Shader and Michael Young},
journal= {arXiv preprint arXiv:1708.00064},
year = {2017}
}