Constructing cospectral graphs via regular rational orthogonal matrix with level two and three

TL;DR

This paper introduces new methods for constructing cospectral graphs using regular rational orthogonal matrices with levels two and three, extending existing switching techniques like GM-switching.

Contribution

It presents two algorithms to identify adjacency matrices compatible with these matrices and introduces generalized switching methods for creating cospectral graphs.

Findings

Developed algorithms for adjacency matrix characterization

Extended GM-switching to new classes of cospectral graphs

Provided theoretical framework for regular rational orthogonal matrices

Abstract

Two graphs $G$ and $H$ are \emph{cospectral} if the adjacency matrices share the same spectrum. Constructing cospectral non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature, e.g. the famous GM-switching method. In this paper, we shall construct cospectral graphs via regular rational orthogonal matrix $Q$ with level two and three. We provide two straightforward algorithms to characterize with adjacency matrix $A$ of graph $G$ such that $Q^TAQ$ is again a (0,1)-matrix, and introduce two new switching methods to construct families of cospectral graphs which generalized the GM-switching to some extent.