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

A signed graph \(\Sigma\) is an ordered pair (G,\(\sigma\)) that consists of a underlying graph \(G=(V,E)\) and a sign mapping called signature \(\sigma\) 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 \(\Sigma\). We then present the adjacency spectrum and Laplacian spectrum of the Cartesian product in terms of the spectrum and Laplacian spectrum of \(\Sigma\), 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.