Lower bounds of the skew spectral radii and skew energy of oriented graphs

Abstract

Let $G$ be a graph with maximum degree $\Delta$, and let $G^{\sigma}$ be an oriented graph of $G$ with skew adjacency matrix $S(G^{\sigma})$. The skew spectral radius $\rho_s(G^{\sigma})$ of $G^\sigma$ is defined as the spectral radius of $S(G^\sigma)$. The skew spectral radius has been studied, but only few results about its lower bound are known. This paper determines some lower bounds of the skew spectral radius, and then studies the oriented graphs whose skew spectral radii attain the lower bound $\sqrt{\Delta}$. Moreover, we apply the skew spectral radius to the skew energy of oriented graphs, which is defined as the sum of the norms of all the eigenvalues of $S(G^\sigma)$, and denoted by $\mathcal{E}_s(G^\sigma)$. As results, we obtain some lower bounds of the skew energy, which improve the known lower bound obtained by Adiga et al.