Some relations between the skew spectrum of an oriented graph and the spectrum of certain closely associated signed graphs

Abstract

Let RG′ be the vertex-edge incidence matrix of an oriented graph G′. Let Λ(Ḟ ) be the signed graph whose vertices are identified as the edges of a signed graph Ḟ , with a pair of vertices being adjacent by a positive (resp. negative) edge if and only if the corresponding edges of Ġ are adjacent and have the same (resp. different) sign. In this paper, we prove that G′ is bipartite if and only if there exists a signed graph Ḟ such that R G′ RG′ − 2I is the adjacency matrix of Λ(Ḟ ). It occurs that Ḟ is fully determined by G′. As an application, in some particular cases we express the skew eigenvalues of G′ in terms of the eigenvalues of Ḟ . We also establish some upper bounds for the skew spectral radius of G′ in both the bipartite and the non-bipartite case.