Relations between the skew spectrum of an oriented graph and the spectrum of an associated signed graph

Abstract

We say that an oriented graph G′=(G,σ′) and a signed graph G˙=(G,σ˙) are mutually associated if σ˙(ik)σ˙(jk)=siksjk holds for every pair of edges ik and jk, where (sij) is the skew adjacency matrix of G′. We prove that this occurs if and only if the underlying graph G is bipartite. On the basis of this result we prove that, in the bipartite case, the skew spectrum of G′ can be obtained from the spectrum of an associated signed graph G˙, and vice versa. In the non-bipartite case, we prove that the skew spectrum of G′ can be obtained from the spectrum of a signed graph associated with the bipartite double of G′. In this way, we show that the theory of skew spectra of oriented graphs has a strong relationship with the theory of spectra of signed graphs. In particular, we demonstrate how some problems concerning oriented graphs can be considered in the framework of signed graphs.