Minimal Skew energy of oriented bicyclic graphs with a given diameter

Abstract

Let S(G σ ) be the skew-adjacency matrix of the oriented graph G σ , which is obtained from a simple undirected graph G by assigning an orientation σ to each of its edges. The skew energy of an oriented graph G σ is defined as the sum of absolute values of all eigenvalues of S(G σ ). For any positive integer d with 3 ≤ d ≤ n − 3, we determine 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.