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arXiv · 2304.06239

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

Shengjie He·Rong-Xia Hao·Hong-Jian Lai·Qiaozhi Geng
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Abstract

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 $c(G)=|E(G)|-|V(G)|+\omega(G)$ 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 $\eta(\widetilde{G})$ be the nullity of $\widetilde{G}$. Chen et al. (2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that $|V(G)|-2m(G)-2c(G) \leq \eta(\widetilde{G}) \leq |V(G)|-2m(G)+2c(G)$, 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 $\eta(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$. 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.

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Shengjie He, Rong-Xia Hao, Hong-Jian Lai, Qiaozhi Geng. 2023-04-13. No mixed graph with the nullity $\eta(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$. https://arxiv.org/abs/2304.06239

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