On the relationship between the skew-rank of an oriented graph and the rank of its underlying graph

An oriented graph Gσ is a digraph without loops and multiple arcs, where G is the underlying graph of Gσ. Let S(Gσ) denote the skew-adjacency matrix of Gσ, and A(G) be the adjacency matrix of G. The rank (resp. skew-rank) of G (resp. Gσ), written as r(G) (resp. sr(Gσ)), refers to the rank of A(G) (resp. S(Gσ)). It is natural and interesting to study the relationship between sr(Gσ) and r(G). Wong, Ma and Tian (European J. Combin. 54 (2016) 76-86) [30] determined the sharp upper bound on sr(Gσ)−r(G). As a continuance of it, in this paper, a sharp lower bound on sr(Gσ)−r(G) is determined; as well a sharp upper bound on sr(Gσ)/r(G) is determined. All the corresponding extremal oriented graphs Gσ are characterized, respectively.