Vertex types in threshold and chain graphs

Abstract

A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. A threshold graph can be obtained from a chain graph by making adjacent all pairs of vertices in one color class. Given a graph G, let λ be an eigenvalue (of the adjacency matrix) of G with multiplicity k ≥ 1. A vertex v of G is a downer, or neutral, or Parter depending whether the multiplicity of λ in G − v is k − 1, or k, or k + 1, respectively. We consider vertex types in the above sense in threshold and chain graphs. In particular, we show that chain graphs can have neutral vertices, disproving a conjecture by Alazemi et al.2000 Mathematics Subject Classification. 05C50.

Tridiagonal matrices and spectral properties of some graph classes

“…Remark 5.2. Taking into account results on the inertia of threshold graphs (for example, see [2], [5]), the results of Theorem 5.1 can be reformulated in the following way:…”