Vertex types in threshold and chain graphs

TL;DR

This paper investigates vertex types in threshold and chain graphs, revealing that chain graphs can have neutral vertices, which disproves a previous conjecture and enhances understanding of their spectral properties.

Contribution

It demonstrates that chain graphs can contain neutral vertices, providing a counterexample to a prior conjecture and deepening spectral graph theory knowledge.

Findings

Chain graphs can have neutral vertices.

Disproved the conjecture by Alazemi et al.

Enhanced understanding of eigenvalue multiplicities in these 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 $\lambda$ 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 $\lambda$ 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.}