On the relationship 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 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 rank (resp. skew-rank) of G (resp. G σ ), written as r ( G ) (resp. s r ( G σ ) ), refers to the rank of A ( G ) (resp. S ( G σ ) ). It is natural and interesting to study the relationship between s r ( G σ ) and r ( G ) . Wong, Ma and Tian (European J. Combin. 54 (2016) 76–86) [30] determined the sharp upper bound on s r ( G σ ) − r ( G ) . As a continuance of it, in this paper, a sharp lower bound on s r ( G σ ) − r ( G ) is determined; as well a sharp upper bound on s r ( G σ ) / r ( G ) is determined. All the corresponding extremal oriented graphs G σ are characterized, respectively. [ABSTRACT FROM AUTHOR]