On the spectrum and energy of Seidel matrix for chain graphs

TL;DR

This paper investigates the spectral properties of the Seidel matrix for connected chain graphs, establishing eigenvalue multiplicities, bounds, and minimal energy configurations, and compares these bounds to existing conjectures.

Contribution

It provides new bounds and exact values for the eigenvalues and energy of the Seidel matrix in chain graphs, improving upon previous conjectures and exploring special cases.

Findings

Eigenvalue -1 always appears in the spectrum.

Eigenvalues other than -1 have multiplicity at most two.

New bounds for Seidel energy outperform existing conjectured bounds.

Abstract

We study various spectral properties of the Seidel matrix $S$ of a connected chain graph. We prove that $-1$ is always an eigenvalue of $S$ and all other eigenvalues of $S$ can have multiplicity at most two. We obtain the multiplicity of the Seidel eigenvalue $-1$, minimum number of distinct eigenvalues, eigenvalue bounds, characteristic polynomial, lower and upper bounds of Seidel energy of a chain graph. It is also shown that the energy bounds obtained here work better than the bounds conjectured by Haemers. We also obtain the minimal Seidel energy for some special chain graphs of order $n$. We also give a number of open problems.