Oriented Bicyclic Graphs with the First Five Large Skew Energies

Abstract

Let $G^{\sigma }$ be an oriented graph obtained by assigning an orientation $\sigma$ to the edge set of a simple undirected graph $G$. Let $S(G^{\sigma })$ be the skew adjacency matrix of $G^{\sigma }$. The skew energy of $G^{\sigma }$ is defined as the sum of the absolute values of all eigenvalues of $S(G^{\sigma })$. In this paper, we give the skew energy order of a family of digraphs and determine the oriented bicyclic graphs of order $n\ge 13$ with the first five largest skew energies, which extends the results of the paper [X. Shen, Y. Hou, C. Zhang, Bicyclic digraphs with extremal skew energy, Electron. J. Linear Algebra 23 (2012) 340-355].