Relation between the skew energy of an oriented graph and its matching number

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 sr(Gσ) of Gσ is the rank of S(Gσ). In this paper, it is proved that ES(Gσ)≥2μ(G) for an arbitrary connected oriented graph Gσ of order n, where μ(G) is the matching number of G, and the equality holds if and only if G is a complete bipartite graph Kn2,n2 with partition (X,Y) of equal size and σ is switching-equivalent to the elementary orientation of G which assigns all edges the same direction from vertices of X to vertices of Y. As an application, we prove that ES(Gσ)≥sr(Gσ) for an oriented graph Gσ and the equality holds if and only if G is the disjoint union of some copies of K2 and some isolated vertices.