Some Relations Between the Skew Spectrum of an Oriented Graph and the Spectrum of Certain Closely Associated Signed Graphs

Abstract

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