Skew-rank of an oriented graph in terms of matching number

An oriented graph GσGσ is a digraph without loops and multiple arcs, where G   is called the underlying graph of GσGσ. Let S(Gσ)S(Gσ) denote the skew-adjacency matrix of GσGσ. The rank of S(Gσ)S(Gσ) is called the skew-rank of GσGσ, denoted by sr(Gσ)sr(Gσ), which is even since S(Gσ)S(Gσ) is skew symmetric. Li and Yu (2015) [12] proved that the skew-rank of an oriented unicyclic graph GσGσ is either 2m(G)−22m(G)−2 or 2m(G)2m(G), where m(G)m(G) denotes the matching number of G  . In this paper, we extend this result to general cases. It is proved that the skew-rank of an oriented connected graph GσGσ is an even integer satisfying 2m(G)−2β(G)≤sr(Gσ)≤2m(G)2m(G)−2β(G)≤sr(Gσ)≤2m(G), where β(G)=|E(G)|−|V(G)|+1β(G)=|E(G)|−|V(G)|+1 is the number of fundamental cycles (also called the first Betti number). Besides, the oriented graphs satisfying sr(Gσ)=2m(G)−2β(G)sr(Gσ)=2m(G)−2β(G) are characterized definitely.