Improved bounds on the $H$-rank of a mixed graph in terms of the matching number and fractional matching number

Abstract

A mixed graph $\widetilde{G}$ is obtained by orienting some edges of a graph $G$, where $G$ is the underlying graph of $\widetilde{G}$. Let $r(\widetilde{G})$ be the $H$-rank of $\widetilde{G}$. Denote by $r(G)$, $\kappa(G)$, $m(G)$ and $m^{\ast}(G)$ the rank, the number of even cycles, the matching number and the fractional matching number of $G$, respectively. Zhou et al. [Discrete Appl. Math. 313 (2022)] proved that $2m(G)-2\kappa(G)\leq r(G)\leq 2m(G)+\rho(G)$, where $\rho(G)$ is the largest number of disjoint odd cycles in $G$. We extend their results to the setting of mixed graphs and prove that $2m(G)-2\kappa(G)\leq r(\widetilde{G}) \leq 2m^{\ast}(G)$ for a mixed graph $\widetilde{G}$. Furthermore, we characterize some classes of mixed graphs with rank $r(\widetilde{G})=2m(G)-2\kappa(G)$, $r(\widetilde{G})=2m(G)-2\kappa(G)+1$ and $r(\widetilde{G})=2m^{\ast}(G)$, respectively. Our results also improve those of Chen et al. [Linear Multiliear Algebra. 66 (2018)]. In addition, our results can be applied to signed graphs and oriented graphs in some situations.