Skew spectra of oriented bipartite graphs

A graph G is said to have a parity-linked orientation phi if every even cycle C-2k in G(phi) is evenly (resp.oddly) oriented whenever k is even (resp.odd).In this paper, this concept is used to provide an affirmative answer to the following conjecture of D.Cui and Y. Hou [D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, The Electronic J. Combin. 20(2):#P19, 2013]: Let G = G(X, Y)be a bipartite graph. Call the X -> Y orientation of G, the canonical orientation. Let phi be any orientation of G and let Sp(S)(G(phi)) and Sp(G) denote respectively the skew spectrum of G(phi) and the spectrum of G. Then Sp(S)(G(phi)) = iSp(G) if and only if phi is switching-equivalent to the canonical orientation of G. Using this result, we determine the switch for a special family of oriented hypercubes Q(d)(phi), d >= 1. Moreover, we give an orientation of the Cartesian product of a bipartite graph and a graph, and then determine the skew spectrum of the resulting oriented product graph, which generalizes a result of Cui and Hou. Further this can be used to construct new families of oriented graphs with maximum skew energy.