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 \( \lambda_1, \lambda_2, \ldots, \lambda_n \) 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 \( m \) (\( n \leq m \leq 2(n-2) \)) 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 \( m \) (\( n \leq m < 2(n-2) \)) arcs. In particular, when the oriented graphs are unicyclic digraphs or bicyclic digraphs, the second and the third minimal skew energy is determined.