不存在零度为 $η(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$ 的混合图
No mixed graph with the nullity $η(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$
AI总结：
本文证明不存在零度恰为顶点数减二倍匹配数加二倍圈数再减一的混合图，并证明对固定圈数存在无穷多个连通混合图达到其余允许零度值。
摘要：
混合图 $\widetilde{G}$ 是通过对其基础简单无向图 $G$ 的某些边进行定向而得到的。设 $c(G)=|E(G)|-|V(G)|+ω(G)$ 为 $G$ 的圈数，其中 $ω(G)$ 为 $G$ 的连通分支数，$m(G)$ 为 $G$ 的匹配数，$η(\widetilde{G})$ 为 $\widetilde{G}$ 的零度。Chen 等人（2018）\cite{LSC} 与 Tian 等人（2018）\cite{TFL} 分别独立证明了 $|V(G)|-2m(G)-2c(G) \leq η(\widetilde{G}) \leq |V(G)|-2m(G)+2c(G)$，并刻画了零度达到上界和下界的混合图。本文证明不存在零度为 $η(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$ 的混合图。此外，对于固定的 $c(G)$，证明了存在无穷多个连通混合图，其零度为 $|V(G)|-2m(G)+2c(G)-s$（$0 \leq s \leq 3c(G), s\neq1$）。
英文摘要：
A mixed graph $\widetilde{G}$ is obtained from a simple undirected graph $G$, the underlying graph of $\widetilde{G}$, by orienting some edges of $G$. Let $c(G)=|E(G)|-|V(G)|+ω(G)$ be the cyclomatic number of $G$ with $ω(G)$ the number of connected components of $G$, $m(G)$ be the matching number of $G$, and $η(\widetilde{G})$ be the nullity of $\widetilde{G}$. Chen et al. (2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that $|V(G)|-2m(G)-2c(G) \leq η(\widetilde{G}) \leq |V(G)|-2m(G)+2c(G)$, 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 $η(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$. Moreover, for fixed $c(G)$, there are infinitely many connected mixed graphs with nullity $|V(G)|-2m(G)+2c(G)-s$ $( 0 \leq s \leq 3c(G), s\neq1 )$ is proved.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
