More on the skew-spectra of bipartite graphs and Cartesian products of graphs

Abstract

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 $S{p}_S(G^{\sigma })$. It is known that a graph $G$ is bipartite if and only if there is an orientation $\sigma$ of $G$ such that $S{p}_S(G^{\sigma })=iSp(G)$. 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.

9 pages

References in corpus (2)

• A survey on the skew energy of oriented graphs
• Oriented unicyclic graphs with extremal skew energy