Some Notes on Spectra of Cographs

Abstract

A cograph is a $P_4$-free graph. We first give a short proof of the fact that $0$ ($-1$) belongs to the spectrum of a connected cograph (with at least two vertices) if and only if it contains duplicate (resp. coduplicate) vertices. As a consequence, we next prove that the polynomial reconstruction of graphs whose vertex-deleted subgraphs have the second largest eigenvalue not exceeding $\frac{\sqrt{5}-1}{2}$ is unique.