No mixed graph with the nullity $\eta (\tilde{G})=|V(G)|-2m(G)+2c(G)-1$

Abstract

A mixed graph $\tilde{G}$ is obtained from a simple undirected graph $G$, the underlying graph of $\tilde{G}$, by orienting some edges of $G$. Let $c(G)=|E(G)|-|V(G)|+\omega (G)$ be the cyclomatic number of $G$ with $\omega (G)$ the number of connected components of $G$, $m(G)$ be the matching number of $G$, and $\eta (\tilde{G})$ be the nullity of $\tilde{G}$. Chen et al. (2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that $|V(G)|-2m(G)-2c(G)\le \eta (\tilde{G})\le |V(G)|-2m(G)+2c(G)$, respectively, and they characterized the mixed graphs with nullity attaining the upper bound and the lower bound. In this paper, we prove that there is no mixed graph with nullity $\eta (\tilde{G})=|V(G)|-2m(G)+2c(G)-1$. Moreover, for fixed $c(G)$, there are infinitely many connected mixed graphs with nullity $|V(G)|-2m(G)+2c(G)-s$ $(0\le s\le 3c(G),s\ne 1)$ is proved.