The maximum multiplicity of an eigenvalue of symmetric matrices with a given graph

TL;DR

This paper introduces two combinatorial parameters, T^-(G) and T^+(G), that precisely bound the maximum eigenvalue multiplicity of symmetric matrices associated with a graph G, enhancing understanding of spectral graph properties.

Contribution

The paper defines and proves the sharpness of two new graph parameters, T^-(G) and T^+(G), that bound the maximum eigenvalue multiplicity for symmetric matrices with a given graph pattern.

Findings

T^-(G) provides a lower bound for M(G).

T^+(G) provides an upper bound for M(G).

Bounds are shown to be sharp.

Abstract

For a graph G, M(G) denotes the maximum multiplicity occurring of an eigenvalue of a symmetric matrix whose zero-nonzero pattern is given by edges of G. We introduce two combinatorial graph parameters T^-(G) and T^+(G) that give a lower and an upper bound for M(G) respectively, and we show that these bounds are sharp.