4-Regular oriented graphs with optimum skew energy

Let G   be a simple undirected graph, and GσGσ be an oriented graph of G with the orientation σ   and skew-adjacency matrix S(Gσ)S(Gσ). The skew energy of the oriented graph GσGσ, denoted by ES(Gσ)ES(Gσ), is defined as the sum of the absolute values of all the eigenvalues of S(Gσ)S(Gσ). In this paper, we characterize the underlying graphs of all 4-regular oriented graphs with optimum skew energy and give orientations of these underlying graphs such that the skew energy of the resultant oriented graphs indeed attain optimum. It should be pointed out that there are infinitely many 4-regular connected optimum skew energy oriented graphs, while the 3-regular case only has two graphs: K4K4 the complete graph on 4 vertices and Q3Q3 the hypercube.