Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph

Let $G^{\sigma}$ be an oriented graph and $S(G^{\sigma})$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^{\sigma}$. The skew-rank of $G^{\sigma}$, denoted by $sr(G^{\sigma})$, is the rank of $S(G^{\sigma})$. Denote by $d(G)=|E(G)|-|V(G)|+\theta(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $\theta(G)$ are the edge number, vertex number and the number of connected components of $G$, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76--86] proved that $sr(G^{\sigma})\leq r(G)+2d(G)$ for an oriented graph $G^{\sigma}$, where $r(G)$ is the rank of the adjacency matrix of $G$, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of $sr(G^{\sigma})$ of an oriented graph $G^{\sigma}$ in terms of $r(G)$ and $d(G)$ of its underlying graph $G$ is left open till now. In this paper, we prove that $sr(G^{\sigma})\geq r(G)-2d(G)$ for an oriented graph $G^{\sigma}$ and characterize the graphs whose skew-rank attain the lower bound.