Oriented Graphs with Minimal Skew Energy

Abstract

The concept of the skew energy of a digraph was introduced by Adiga, Balakrishnan and So in 2010. An oriented graph $G^{\sigma }$ is a simple undirected graph $G$ with an orientation, which assigns to each edge a direction so that $G^{\sigma }$ becomes a directed graph. Then $G$ is called the underlying graph of $G^{\sigma }$. Let $S(G^{\sigma })$ be the skew-adjacency matrix of $G^{\sigma }$ and ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ denote all the eigenvalues of $S(G^{\sigma })$. The skew energy of $G^{\sigma }$ is defined as the sum of the absolute values of all eigenvalues of $S(G^{\sigma })$. Recently, Gong, Li and Xu determined all oriented graphs with minimal skew energy among all connected oriented graphs on $n$ vertices with $m$ ($n\le m\le 2(n-2)$) arcs. In this paper, we determine all oriented graphs with the second and the third minimal skew energy among all connected oriented graphs with $n$ vertices and $m$ ($n\le m<2(n-2)$) arcs. In particular, when the oriented graphs are unicyclic digraphs or bicyclic digraphs, the second and the third minimal skew energy is determined.