文摘
An oriented graph Gσ is a digraph without loops and multiple arcs, where G is called the underlying graph of Gσ. Let S(Gσ) denote the skew-adjacency matrix of Gσ. The rank of S(Gσ) is called the skew-rank of Gσ, denoted by 58ca32f3ed445d8cf6f055ade2" title="Click to view the MathML source">sr(Gσ), which is even since S(Gσ) is skew symmetric. Li and Yu (2015) [12] proved that the skew-rank of an oriented unicyclic graph Gσ 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 Gσ is an even integer satisfying 2m(G)−2β(G)≤sr(Gσ)≤2m(G), where β(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) are characterized definitely.