Skew spectra of oriented bipartite graphs

Abstract

A graph G is said to have a parity-linked orientation φ if every even cycle C2k in Gπ 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 Φ 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 Q*, 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.