On the skew spectra of Cartesian products of graphs

An oriented graph G(sigma) is a simple undirected graph G with an orientation sigma, which assigns to each edge of G a direction so that G(sigma) becomes a directed graph. G is called the underlying graph of G(sigma) and we denote by S (G(sigma)) the skew-adjacency matrix of G(sigma) and its spectrum Sp(G(sigma)) is called the skew-spectrum of G(sigma). In this paper, the skew spectra of two orientations of the Cartesian product of two graphs are discussed. As applications, new families of oriented bipartite graphs G(sigma) with Sp(G(sigma)) - iSp(G) are given and the orientation of a product graph with maximum skew energy is obtained.