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. 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.

The micro-world of cographs

“…its collection M (κ) of minimal hereditary subclasses of cographs where κ is unbounded, and the parameters are compared by their strength. There are many other interesting parameters that are unbounded in the class of cographs, such as Dilworth number [29], distinguishing number [9], shrub-depth [28], rank [15], metric dimension [46], etc. However, surprisingly, there are not so many "interesting" subclasses of cographs that appear in the characterisation of those parameters.…”

Characterizing and computing weight-equitable partitions of graphs

“…From the spectral point of view, Jung [21] introduced an algorithm for locating eigenvalues of cographs in a given interval. Ghorbani [14] provided a new characterization of cographs, and further properties of the eigenvalues (of the adjacency matrix) of a cograph were explored, e.g., by Ghorbani [13], Mohammadian and Trevisan [23], and Jacobs et [20].…”