Distance eigenvalues of a cograph and their multiplicities

Abstract

A cograph is a graph with no induced path on four vertices. The vicinal preorder on the vertex set of a graph is defined in terms of inclusions among the neighborhoods of vertices. The minimum number of chains with respect to the vicinal preorder required to cover the vertex set of a graph G is called the Dilworth number of G. In this paper, we prove that for a connected cograph G, the multiplicity of any distance eigenvalue except −2 and −1 is at most the Dilworth number of G. Furthermore, we show that no connected cographs have distance eigenvalues in the interval (−2,−1), which generalizes a result of Lu et al. (2018) [24].