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

Let $G^σ$ be an oriented graph and $S(G^σ)$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^σ$. The skew-rank of $G^σ$, denoted by $sr(G^σ)$, is the rank of $S(G^σ)$. Denote by $d(G)=|E(G)|-|V(G)|+θ(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $θ(G)$ 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 $sr(G^σ)\leq r(G)+2d(G)$ for an oriented graph $G^σ$, 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 $sr(G^σ)$ of an oriented graph $G^σ$ in terms of $r(G)$ and $d(G)$ of its underlying graph $G$ is left open till now. In this paper, we prove that $sr(G^σ)\geq r(G)-2d(G)$ for an oriented graph $G^σ$ and characterize the graphs whose skew-rank attain the lower bound.