A note on the minimum skew rank of a graph

The minimum skew rank $mr^{-}(\mathbb{F},G)$ of a graph $G$ over a field $\mathbb{F}$ is the smallest possible rank among all skew symmetric matrices over $\mathbb{F}$, whose ($i$,$j$)-entry (for $i\neq j$) is nonzero whenever $ij$ is an edge in $G$ and is zero otherwise. We give some new properties of the minimum skew rank of a graph, including a characterization of the graphs $G$ with cut vertices over the infinite field $\mathbb{F}$ such that $mr^{-}(\mathbb{F},G)=4$, determination of the minimum skew rank of $k$-paths over a field $\mathbb{F}$, and an extending of an existing result to show that $mr^{-}(\mathbb{F},G)=2match(G)=MR^{-}(\mathbb{F},G)$ for a connected graph $G$ with no even cycles and a field $\mathbb{F}$, where $match(G)$ is the matching number of $G$, and $MR^{-}(\mathbb{F},G)$ is the largest possible rank among all skew symmetric matrices over $\mathbb{F}$.