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

TL;DR

This paper proves a conjecture about the uniqueness of certain orientations of bipartite graphs related to their skew-spectrum and extends the analysis to Cartesian product graphs, providing new methods to construct graphs with maximum skew energy.

Contribution

It proves the conjecture that the orientation of bipartite graphs with a specific skew-spectrum is unique under switching-equivalence and generalizes the skew-spectrum analysis to Cartesian product graphs.

Findings

Confirmed the uniqueness of the orientation for bipartite graphs with a given skew-spectrum.

Derived the skew-spectrum of Cartesian product graphs involving bipartite graphs.

Provided a method to construct oriented graphs with maximum skew energy.

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 $Sp_S(G^\sigma)$. 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.