Skew Spectra of Oriented Graphs

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