On adjacency and Laplacian cospectral non-isomorphic signed graphs

TL;DR

This paper introduces new operations on signed graphs that generate non-isomorphic graphs sharing the same Laplacian spectrum, advancing understanding of spectral graph theory and providing methods to construct cospectral non-isomorphic signed graphs.

Contribution

It presents two novel operations on signed graphs that relate adjacency and Laplacian spectra, enabling the construction of cospectral non-isomorphic signed graphs and integral signed graphs.

Findings

Established relationships between adjacency and Laplacian spectra of signed graphs.

Constructed multiple pairs of cospectral non-isomorphic signed graphs.

Developed methods to generate integral signed graphs.

Abstract

Let $\Gamma=(G,\sigma)$ be a signed graph, where $\sigma$ is the sign function on the edges of $G$. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs.