On the minimum number of eigenvalues of matrices associated with cographs

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.

💡 Research Summary

The paper investigates the parameter q(G), defined as the smallest possible number of distinct eigenvalues among all real symmetric matrices whose off‑diagonal zero–nonzero pattern exactly matches the adjacency of a given graph G. Formally, for a graph G with vertex set V={v₁,…,vₙ}, the set S(G) consists of all symmetric matrices M∈ℝⁿˣⁿ such that for i≠j, m_{ij}≠0 iff {v_i,v_j}∈E(G). The diagonal entries are unrestricted. The goal is to determine q(G)=min_{M∈S(G)}|DSpec(M)|, where DSpec(M) denotes the set of distinct eigenvalues of M.

Previous work has examined q(G) for trees, where a lower bound q(T)≥diam(T)+1 holds, and for threshold graphs—a subclass of cographs—where it was shown that q(G)≤4. The present work extends this bound to the entire class of cographs (complement‑reducible graphs), which are precisely the graphs that contain no induced P₄ (a path on four vertices). Cographs admit a recursive construction using disjoint unions (∪) and joins (⊕), and can be uniquely represented by a cotree in normalized form: a rooted tree whose leaves correspond to vertices and internal nodes are labeled either ∪ or ⊕, alternating along any root‑to‑leaf path.

The main result (Theorem 2) states: for any cograph G and any non‑zero real number λ, there exists a matrix M∈S(G) such that DSpec(M)⊆{−λ, 0, λ, 2λ}. Consequently, q(G)≤4 for all cographs. The proof proceeds by induction on the number of vertices n.

Base case n=1 is trivial (M=