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

Abstract

Let Gσ be an oriented graph with skew adjacency matrix S(Gσ). The skew energy Es(Gσ) of Gσ is the sum of the norms of all eigenvalues of S(Gσ) and the skew rank rs(Gσ) of Gσ is the rank of S(Gσ). Recently, Tian and Wong gave a lower bound of Es(Gσ) in terms of its skew rank. They proved that rs(Gσ)≤Es(Gσ) 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 thatEs(Gσ)≤rs(Gσ)Δ for an arbitrary oriented graph Gσ with maximum degree Δ, and the upper bound is attained if and only if Gσ is the disjoint union of rs(Gσ)2 copies of (KΔ,Δ)σ together with some isolated vertices, where the orientation σ is switching-equivalent to the elementary orientation of KΔ,Δ which assigns all edges the same direction from a chromatic set to another one.