Relation between the skew-rank of an oriented graph and the rank of its underlying graph.

An oriented graph G σ is a digraph without loops and multiple arcs, where G is called the underlying graph of G σ . Let S ( G σ ) denote the skew-adjacency matrix of G σ , and A ( G ) be the adjacency matrix of G . The skew-rank of G σ , written as s r ( G σ ) , refers to the rank of S ( G σ ) , which is always even since S ( G σ ) is skew symmetric. A natural problem is: How about the relation between the skew-rank of an oriented graph G σ and the rank of its underlying graph? In this paper, we focus our attention on this problem. Denote by d ( G ) the dimension of cycle spaces of G , that is d ( G ) = | E ( G ) | − | V ( G ) | + θ ( G ) , where θ ( G ) denotes the number of connected components of G . It is proved that s r ( G σ ) ≤ r ( G ) + 2 d ( G ) for an oriented graph G σ , the oriented graphs G σ whose skew-rank attains the upper bound are characterized. [ABSTRACT FROM AUTHOR]