arXiv 1909.075552019-09math.CO

复数单位增益图的匹配数秩

The rank of a complex unit gain graph in terms of the matching number

Shengjie He, Rong-Xia Hao, Fengming Dong

摘要：

复单位增益图（或${\mathbb T}$ -增益图）是由作为$(G, φ)$的基础图的简单图$G$、单位复数集合$\mathbb{T}= \{ z \in C:|z|=1 \}$和具有以下性质的增益函数$φ: \overrightarrow{E} \rightarrow \mathbb{T}$组成的三元组$Φ=(G, {\mathbb T}, φ)$（或简称$(G, φ)$）：$φ(e_{i,j})=φ(e_{j,i})^{-1}$。本文证明了$2m(G)-2c(G) \leq r(G, φ) \leq 2m(G)+c(G)$，其中$r(G, φ)$，$m(G)$和$c(G)$分别是Hermite邻接矩阵$H(G, φ)$的秩，$G$的匹配数和圈数.此外，还刻画了具有$r(G, φ)=2m(G)-2c(G)$和$r(G, φ)=2m(G)+c(G)$的复单位增益图$(G, \mathbb{T}, φ)$。这些结果推广了无向图、混合图和符号图的相应结果。此外，我们表明，$2m(G-V_{0}) \leq r(G, φ) \leq 2m(G)+b(G)$持有$V(G)$的任何子集$V_0$，使$G-V_0$是无环和$b(G)$是最小整数$|S|$，使$G-S$是二分的$S \subset V(G)$。

英文摘要：

A complex unit gain graph (or ${\mathbb T}$-gain graph) is a triple $Φ=(G, {\mathbb T}, φ)$ (or $(G, φ)$ for short) consisting of a simple graph $G$, as the underlying graph of $(G, φ)$, the set of unit complex numbers $\mathbb{T}= \{ z \in C:|z|=1 \}$ and a gain function $φ: \overrightarrow{E} \rightarrow \mathbb{T}$ with the property that $φ(e_{i,j})=φ(e_{j,i})^{-1}$. In this paper, we prove that $2m(G)-2c(G) \leq r(G, φ) \leq 2m(G)+c(G)$, where $r(G, φ)$, $m(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $H(G, φ)$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, the complex unit gain graphs $(G, \mathbb{T}, φ)$ with $r(G, φ)=2m(G)-2c(G)$ and $r(G, φ)=2m(G)+c(G)$ are characterized. These results generalize the corresponding known results about undirected graphs, mixed graphs and signed graphs. Moreover, we show that $2m(G-V_{0}) \leq r(G, φ) \leq 2m(G)+b(G)$ holds for any subset $V_0$ of $V(G)$ such that $G-V_0$ is acyclic and $b(G)$ is the minimum integer $|S|$ such that $G-S$ is bipartite for $S \subset V(G)$.

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。

摘要：

英文摘要：

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