The skew energy of a digraph

We are interested in the energy of the skew-adjacency matrix of a directed graph D, which is simply called the skew energy of D in this paper. Properties of the skew energy of D are studied. In particular, a sharp upper bound for the skew energy of D is derived in terms of the order of D and the maximum degree of its underlying undirected graph. An infinite family of digraphs attaining the maximum skew energy is constructed. Moreover, the skew energy of a directed tree is independent of its orientation, and interestingly it is equal to the energy of the underlying undirected tree. Skew energies of directed cycles under different orientations are also computed. Some open problems are presented.