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.