Inverse eigenvalue and related problems for hollow matrices described by graphs

TL;DR

This paper investigates the spectra of hollow matrices associated with graphs, solving inverse eigenvalue problems for specific graph families and exploring related spectral properties.

Contribution

It provides solutions to the hollow inverse eigenvalue problem for paths and bipartite graphs, and analyzes spectral properties for other graph families.

Findings

Solutions for paths and bipartite graphs are presented.

Results on eigenvalue multiplicities and spectral diversity are provided.

Spectral properties are characterized for various graph classes.

Abstract

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.