The inverse eigenvalue problem of a graph: Multiplicities and minors

TL;DR

This paper extends the understanding of the inverse eigenvalue problem for graphs by exploring minor monotonicity and applying these concepts to solve the problem for all graphs of order five and to identify forbidden minors.

Contribution

It introduces a form of minor monotonicity for the inverse eigenvalue problem and applies it to classify graphs of order five and forbidden minors with limited eigenvalue multiplicities.

Findings

Extended the inverse eigenvalue problem to minor monotonicity with restrictions.

Solved the inverse eigenvalue problem for all graphs of order five.

Characterized forbidden minors for graphs with at most one multiple eigenvalue.

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.