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

Abstract

Let $R_{G'}$ be the vertex-edge incidence matrix of an oriented graph $G'$, and let $\Lambda(\dot{F})$ be the adjacency matrix of the signed graph whose vertices are identified as the edges of a signed graph $\dot{F}$ and a pair of vertices is adjacent by a positive (resp. negative) edge if and only if the corresponding edges of $\dot{G}$ have the same (resp. different) sign. In this paper, we prove that $G'$ is bipartite if and only if there exists a signed graph $\dot{F}$, such that $R_{G'}^\intercal R_{G'}-2I$ is the adjacency matrix of $\Lambda(\dot{F})$. It occurs that $\dot{F}$ 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 $\dot{F}$. We also establish some upper bounds for the skew spectral radius of $G'$ in both bipartite and non-bipartite case.