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 \(n \geq 13\) 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].