The maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph

TL;DR

This paper introduces a new graph parameter $Mm(G)$ measuring the maximum minimal eigenvalue multiplicity of symmetric matrices constrained by a graph, and computes it for various graph families.

Contribution

It defines the parameter $Mm(G)$ and provides explicit calculations for several classes of graphs, advancing understanding of eigenvalue multiplicities in graph-constrained matrices.

Findings

$Mm(G)$ computed for several graph families

Provides bounds and exact values for specific graphs

Enhances understanding of eigenvalue multiplicities in graph matrices

Abstract

In this paper we introduce a parameter $Mm(G)$, defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph $G$. We compute $Mm(G)$ for several families of graphs.