Skew spectra of oriented graphs

Abstract

An oriented graph Gσ is a simple undirected graph G with an orientation σ, which assigns to each edge a direction so that Gσ becomes a directed graph. G is called the underlying graph of Gσ, and we denote by Sp(G) the adjacency spectrum of G. Skew-adjacency matrix S(Gσ) of Gσ is introduced, and its spectrum SpS(Gσ) is called the skew-spectrum of Gσ. The relationship between SpS(G σ) and Sp(G) is studied. In particular, we prove that (i) SpS(Gσ) = iSp(G) for some orientation σ if and only if G is bipartite, (ii) SpS(Gσ) = iSp(G) for any orientation σ if and only if G is a forest, where i = √-1.