Some spectral properties of chain graphs

TL;DR

This paper investigates the spectral properties of chain graphs, disproves a conjecture about shared eigenvalues with their subgraphs, and provides new insights into their eigenvalue intervals.

Contribution

The paper disproves a conjecture regarding eigenvalues of chain graphs and their vertex-deleted subgraphs, and establishes new spectral properties of chain graphs.

Findings

Disproved the conjecture that no chain graph shares a non-zero eigenvalue with its vertex-deleted subgraphs.

Showed the conjecture holds for subgraphs obtained by deleting vertices of maximum degree.

Proved chain graphs have no eigenvalues in the interval (0, 1/2).

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. Alazemi, Andeli\'c and Simi\'c conjectured that no chain graph shares a non-zero (adjacency) eigenvalue with its vertex-deleted subgraphs. We disprove this conjecture. However, we show that the assertion holds for subgraphs obtained by deleting vertices of maximum degrees in either of color classes. We also give a simple proof for the fact that chain graphs have no eigenvalue in the interval $(0,1/2)$.