# Constructing cospectral signed graphs

## TL;DR

This paper extends spectral graph theory techniques to signed graphs, demonstrating how to construct pairs of cospectral, nonisomorphic signed graphs using adapted routines originally designed for unsigned graphs.

## Contribution

It adapts Godsil-McKay routines for signed graphs, enabling the construction of cospectral, nonisomorphic signed graphs, a novel extension in spectral graph theory.

## Key findings

- Successfully adapted routines for signed graphs.
- Constructed pairs of cospectral, nonisomorphic signed graphs.
- Extended understanding of spectral properties in signed graphs.

## Abstract

A well--known fact in Spectral Graph Theory is the existence of pairs of isospectral nonisomorphic graphs (known as PINGS). The work of A.J. Schwenk (in 1973) and of C. Godsil and B. McKay (in 1982) shed some light on the explanation of the presence of isospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil-McKay--type routines developed for graphs, whose adjacency matrices are $(0,1)$-matrices, to the level of signed graphs, whose adjacency matrices allow the presence of $-1$'s. We show that, with suitable adaption, such routines can be successfully ported to signed graphs, and we can build pairs of cospectral switching nonisomorphic signed graphs.