  Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the
orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum
of $S(G^\sigma)$ is called the skew-spectrum of $G^\sigma$, denoted by
$Sp_S(G^\sigma)$. It is known that a graph $G$ is bipartite if and only if
there is an orientation $\sigma$ of $G$ such that $Sp_S(G^\sigma)=iSp(G)$. In
[D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, Electron.
J. Combin. 20(2013), #P19], Cui and Hou conjectured that such orientation of a
bipartite graph is unique under switching-equivalence. In this paper, we prove
that the conjecture is true. 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 Cui and Hou's
result, and can be used to construct more oriented graphs with maximum skew
energy.
