Skew Spectra of Oriented Bipartite Graphs

A graph G is said to have a parity-linked orientation φ if every even cycle C2k in G 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 φ be any orientation of G and let SpS(G ) and Sp(G) denote respectively the skew spectrum of G and the spectrum of G. Then SpS(G ) = iSp(G) if and only if φ is switching-equivalent to the canonical orientation of G. Using this result, we determine the switch for a special family of oriented hypercubes Qφd , 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.