Cographs: Eigenvalues and Dilworth number

Abstract

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. 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. We prove that for any cograph G, the multiplicity of any eigenvalue λ≠0,−1, does not exceed the Dilworth number of G and show that this bound is tight. Royle (2003) proved that if a cograph G has no pair of vertices with the same neighborhood, then G has no 0 eigenvalue, and asked if besides cographs, there are any other natural classes of graphs for which this property holds. We give a partial answer to this question by showing that an H-free family of graphs has this property if and only if it is a subclass of the family of cographs. A similar result is also shown to hold for the −1 eigenvalue.