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

An oriented graph G(sigma) is a digraph without loops and multiple arcs, where G is called the underlying graph of G(sigma). Let S(G(sigma)) denote the skew-adjacency matrix of G(sigma). The rank of S(G(sigma)) is called the skew-rank of G(sigma), denoted by sr(G(sigma)), which is even since S(G(sigma)) is skew symmetric. Li and Yu (2015) [12] proved that the skew-rank of an oriented unicyclic graph G(sigma) is either 2m(G) - 2 or 2m(G), where 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 G(sigma) is an even integer satisfying 2m(G) - 2 beta(G) <= sr(G(sigma)) <= 2m(G), where beta(G) = vertical bar E(G)vertical bar - vertical bar V (G)vertical bar + 1 is the number of fundamental cycles (also called the first Betti number). Besides, the oriented graphs satisfying sr(G(sigma)) = 2m(G) - 2 beta(G) are characterized definitely. (C) 2016 Elsevier Inc. All rights reserved.