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

TL;DR

This paper extends bounds on the $H$-rank of mixed graphs using matching and fractional matching numbers, providing new characterizations and improving previous results, with applications to signed and oriented graphs.

Contribution

It introduces new bounds for the $H$-rank of mixed graphs based on matching parameters and characterizes classes of graphs achieving these bounds.

Findings

Established bounds: $2m(G)-2\kappa(G) \\leq r( ilde{G}) \\leq 2m^*(G)$.

Characterized classes of mixed graphs with specific $H$-rank values.

Improved previous bounds and extended applicability to signed and oriented graphs.

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.