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

2013/05/15 by Chen, Xiaolin, Li, Xueliang, Lian, Huishu
#05C20 #05C50 #05C90 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1305.3414

Abstract

Given a graph G, let G^σ be an oriented graph of G with the orientation σ and skew-adjacency matrix S(G^σ). Then the spectrum of S(G^σ) is called the skew-spectrum of G^σ, denoted by Sp_S(G^σ). It is known that a graph G is bipartite if and only if there is an orientation σ of G such that Sp_S(G^σ)=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.

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