On the Skew Spectra of Cartesian Products of Graphs

An oriented graph ${G^{\sigma}}$ is a simple undirected graph $G$ with an orientation, 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 products are discussed, as applications, new families of oriented bipartite graphs ${G^{\sigma}}$ with $Sp({G^{\sigma}})={\bf i} Sp(G)$ are given and the orientation of a product graph with maximum skew energy is obtained.