Cographs: Eigenvalues and Dilworth Number

TL;DR

This paper investigates the spectral properties of cographs, establishing bounds on eigenvalue multiplicities related to the Dilworth number and characterizing classes of graphs with similar eigenvalue properties.

Contribution

It proves that for cographs, the multiplicity of non-zero, non-minus-one eigenvalues is bounded by the Dilworth number and characterizes graph classes with no zero or minus-one eigenvalues.

Findings

Eigenvalue multiplicity bounds for cographs are tight.

Characterization of H-free graphs with no zero or minus-one eigenvalues.

Extension of eigenvalue properties beyond cographs.

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 $\lambda\ne0,-1$, does not exceed the Dilworth number of $G$ and show that this bound is tight. G. F. Royle [The rank of a cograph, Electron. J. Combin. 10 (2003), Note 11] proved that if a cograph $G$ has no pair of vertices with the same neighborhood, then $G$ has no 0 eigenvalue, and asked if beside 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.