On the minimum number of eigenvalues of matrices associated with cographs

TL;DR

We study the minimum number of distinct eigenvalues $q(G)$ for real symmetric matrices associated with a graph, proving a sharp spectral bound for the whole class of cographs. The authors constructively extend matrices from smaller cographs via twin operations, using a four-case eigenbasis extension to enforce a restricted spectrum: for any $\lambda\neq0$, there exists $M\in S(G)$ with $\mathrm{DSpec}(M)\subseteq\{-\lambda,0,\lambda,2\lambda\}$, and in particular $q(G)\le 4$. A stronger statement also ensures diagonal entries lie in $\{0,\lambda\}$. The results generalize a known bound for threshold graphs and highlight how cotree/twins structure enables spectral control in cographs, with implications for inverse eigenvalue problems and graph identification.

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.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof