On the Seidel spectrum of threshold graphs

TL;DR

This paper investigates the spectral properties of the Seidel matrix in connected threshold graphs, providing formulas for eigenvalues, characteristic polynomial, and constructing Seidel cospectral graphs.

Contribution

It introduces new formulas for eigenvalues and characteristic polynomial of the Seidel matrix in threshold graphs, and constructs families of Seidel cospectral graphs.

Findings

Formulas for eigenvalues ±1 of the Seidel matrix

Characterization of threshold graphs with up to 5 distinct Seidel eigenvalues

Construction of Seidel cospectral threshold graphs

Abstract

In this paper, we analyse spectral properties of Seidel matrix (denoted by $S$) of connected threshold graphs. We compute the characteristic polynomial and determinant of Seidel matrix of threshold graphs. We derive formulas for the multiplicity of the eigenvalues $\pm 1$ of $S$. Further we determine threshold graphs with at most 5 distinct Seidel eigenvalues. Finally we construct families of Seidel cospectral threshold graphs.