An Alternative Technique to Find the Spectrum and Laplacian Spectrum of a Cartesian Product of Signed Graphs

A signed graph \(\Sigma\) $\mathrm{\Sigma }$ is an ordered pair (G, \(\sigma\) $\sigma$ ) that consists of a underlying graph \(G=(V,E)\) $G=(V,E)$ and a sign mapping called signature \(\sigma\) $\sigma$ from E to the sign set \(\lbrace +, - \rbrace\) $\{+,-\}$ . In this article, we provide another way of looking at the Cartesian product of a path graph and an arbitrary signed graph \(\Sigma\) $\mathrm{\Sigma }$ . We then present the adjacency spectrum and Laplacian spectrum of the Cartesian product in terms of the spectrum and Laplacian spectrum of \(\Sigma\) $\mathrm{\Sigma }$ , respectively. We further provide an upper bound and lower bound for the respective energies. As applications, the results in this article are used (1) to construct a family of infinitely many cospectral and Laplacian cospectral graphs and (2) to compute the adjacency (respectively, Laplacian) spectrum of some known classes of graphs.