A short note on cospectral and integral chain graphs for Seidel matrix

TL;DR

This paper explores cospectral and integral chain graphs related to Seidel matrices, providing formulas for generating infinite inequivalent graphs with the same spectrum and answering a posed problem negatively.

Contribution

It introduces a formula to generate infinitely many inequivalent cospectral chain graphs and constructs Seidel integral chain graphs, addressing a previously open problem.

Findings

Generated infinite inequivalent cospectral chain graphs.

Constructed Seidel integral chain graphs.

Answered a question on regular graphs in Seidel matrix switching classes.

Abstract

In this brief communication, we investigate the cospectral as well integral chain graphs for Seidel matrix, a key component to study the structural properties of equiangular lines in space. We derive a formula that allows to generate an infinite number of inequivalent chain graphs with identical spectrum. In addition, we obtain a family of Seidel integral chain graphs. This contrapositively answers a problem posed by Greaves ["Equiangular line systems and switching classes containing regular graphs", Linear Algebra Appl., (2018)] ("Does every Seidel matrix with precisely three distinct rational eigenvalues contain a regular graph in its switching class?"). Our observation is- "no".