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 $k \geq 1$. 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 {\em et al.}