Bicyclic digraphs with extremal skew energy

Abstract

Abstract. Let − → G be a digraph and S( − → G ) be the skew-adjacency matrix of − → G . The skew energy of − → G is the sum of the absolute values of eigenvalues of S( − → G ). In this paper, the bicyclic digraphs with minimal and maximal skew energy are determined.

Minimal Skew energy of oriented bicyclic graphs with a given diameter

“…For the oriented unicyclic graphs of order n, Hou et al [5] obtained the oriented graphs with the 1st-minimal, the 2nd-minimal and the maximal skew energies, and Zhu [18] determined the oriented graphs with the first ⌊ n−9 2 ⌋ largest skew energies. For the oriented bicyclic graphs, Shen et al [11] deduced the oriented graphs with the minimal and maximal skew energies, and Wang et al [13] characterized the oriented graph with the second largest skew energy. Zhu and Yang [19] obtained the oriented unicyclic graphs that have perfect matchings with the minimal skew energy.…”

“…The graph with minimal skew energy among all oriented bicyclic graphs that contain no vertex disjoint odd cycle of lengths s and l with s + l ≡ 2(mod4) on n vertices with a given diameter d will be determined in Section 3, where 3 ≤ d ≤ n − 3. For d = 2, we can refer to [11].…”

On oriented graphs with minimal skew energy

“…Shen el al. [23] determined the bicyclic graphs with the maximal and minimal energies. Gong and Xu [9] determined the 3-regular graphs with the optimum skew energy, and Tian [24] determined the hypercubes with the optimum skew energy.…”

The skew energy of random oriented graphs

“…Moreover, other bounds and the extremal graphs of some classes of oriented graphs have been established. In [11] and [16], Hou et. al.…”