On the minimum number of eigenvalues of matrices associated with cographs

TL;DR

This paper proves that for any cograph, there exists an associated symmetric matrix with at most four distinct eigenvalues, advancing understanding of spectral properties related to graph structure.

Contribution

It establishes that every cograph admits an associated matrix with no more than four eigenvalues, revealing a spectral bound specific to this class of graphs.

Findings

Existence of an associated matrix with ≤4 eigenvalues for all cographs

Advances spectral graph theory by linking cograph structure to eigenvalue bounds

Provides a new perspective on the spectral properties of cographs

Abstract

A symmetric matrix $M=(m_{ij}) \in \mathbb{R}^{n \times n}$ is said to be associated with an $n$-vertex graph $G=(V,E)$ with vertex set $\{v_1,\ldots,v_n\}$ if, for every $i \neq j$, we have $m_{ij} \neq 0$ if and only if $\{v_i,v_j\}\in E$. We prove that, for every cograph $G$, there is a matrix $M$ associated with $G$ for which the number of distinct eigenvalues is at most 4.