38 research outputs found

    No mixed graph with the nullity Ξ·(G~)=∣V(G)βˆ£βˆ’2m(G)+2c(G)βˆ’1\eta(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1

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    A mixed graph G~\widetilde{G} is obtained from a simple undirected graph GG, the underlying graph of G~\widetilde{G}, by orienting some edges of GG. Let c(G)=∣E(G)βˆ£βˆ’βˆ£V(G)∣+Ο‰(G)c(G)=|E(G)|-|V(G)|+\omega(G) be the cyclomatic number of GG with Ο‰(G)\omega(G) the number of connected components of GG, m(G)m(G) be the matching number of GG, and Ξ·(G~)\eta(\widetilde{G}) be the nullity of G~\widetilde{G}. Chen et al. (2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that ∣V(G)βˆ£βˆ’2m(G)βˆ’2c(G)≀η(G~)β‰€βˆ£V(G)βˆ£βˆ’2m(G)+2c(G)|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 Ξ·(G~)=∣V(G)βˆ£βˆ’2m(G)+2c(G)βˆ’1\eta(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1. Moreover, for fixed c(G)c(G), there are infinitely many connected mixed graphs with nullity ∣V(G)βˆ£βˆ’2m(G)+2c(G)βˆ’s|V(G)|-2m(G)+2c(G)-s (0≀s≀3c(G),sβ‰ 1)( 0 \leq s \leq 3c(G), s\neq1 ) is proved
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