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 $\{v_1,\dots ,v_n\}$ if, for every $i\mathrm{\ne }j$, we have $m_{ij}\mathrm{\ne }0$ 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.