On oriented graphs with minimal skew energy

Abstract

Let S(G^Ï) be the skew-adjacency matrix of an oriented graph GÏ. The skew energy of G^Ï is the sum of all singular values of its skew-adjacency matrix S(G^Ï). This paper first establishes an integral formula for the skew energy of an oriented graph. Then, it determines all oriented graphs with minimal skew energy among all connected oriented graphs on n vertices with m (n ⤠m < 2(n â 2)) arcs, which is analogous to the conjecture for the energy of undirected graphs proposed by Caporossi et al. [G. Caporossi, D. Cvetkovic, I. Gutman, and P. Hansen. Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy. J. Chem. Inf. Comput. Sci., 39:984â996, 1999].