5-regular oriented graphs with optimum skew energy

Abstract

Let $G$ be a simple undirected graph and $G^\sigma$ be the corresponding oriented graph of $G$ with the orientation $\sigma$. The skew energy of $G^\sigma$, denoted by $\varepsilon_s(G^\sigma)$, is defined as the sum of the singular values of the skew adjacency matrix $S(G^\sigma)$. In 2010, Adiga et al. certified that $\varepsilon_s(G^\sigma) \leq n\sqrt{\Delta}$, where $\Delta$ is the maximum degree of $G$ of order $n$. In this paper, we determine all connected 5-regular oriented graphs of order $n$ with maximum skew-energy.