Upper bound of skew energy of an oriented graph in terms of its skew rank

Abstract Let G σ be an oriented graph with skew adjacency matrix S ( G σ ) . The skew energy E s ( G σ ) of G σ is the sum of the norms of all eigenvalues of S ( G σ ) and the skew rank r s ( G σ ) of G σ is the rank of S ( G σ ) . Recently, Tian and Wong gave a lower bound of E s ( G σ ) in terms of its skew rank. They proved that r s ( G σ ) ≤ E s ( G σ ) and they characterized the oriented graphs which satisfy the equality. In this paper, we aim to establish an upper bound for skew energy of an oriented graph in terms of its skew rank and maximum vertex degree. It is proved that E s ( G σ ) ≤ r s ( G σ ) Δ for an arbitrary oriented graph G σ with maximum degree Δ, and the upper bound is attained if and only if G σ is the disjoint union of r s ( G σ ) 2 copies of ( K Δ , Δ ) σ together with some isolated vertices, where the orientation σ is switching-equivalent to the elementary orientation of K Δ , Δ which assigns all edges the same direction from a chromatic set to another one.