用匹配数表示的复单位增益图的惯性指数
Inertia indices of a complex unit gain graph in terms of matching number
摘要：
复数单位增益图是三元$φ=(G, \mathbb{T}, φ)$（或简称$G^φ$），其由作为$G^φ$的基础图的简单图$G$、单位复数集合$\mathbb{T}={z\in \mathbb{C}: |z| = 1}$和增益函数$φ: \overrightarrow{E}\rightarrow \mathbb{T}$组成，使得$φ(e_{i,j})=φ(e_{j,i}) ^{-1}$。设$A(G^φ)$为$G^φ$的邻接矩阵。本文证明了$$m(G)-c(G)\leq p(G^φ)\leq m(G)+c(G),$$ $$m(G)-c(G)\leq n(G^φ)\leq m(G)+c(G),$$，其中$p(G^φ)$，$n(G^φ)$，$m(G)$和$c(G)$分别是$A(G^φ)$的正特征值个数，$A(G^φ)$的负特征值个数，$G$的匹配数和圈数.此外，我们还分别刻画了达到上界和下界的图。
英文摘要：
A complex unit 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 $\mathbb{T}={z\in \mathbb{C}: |z| = 1}$ and a gain function $φ: \overrightarrow{E}\rightarrow \mathbb{T}$ such that $φ(e_{i,j})=φ(e_{j,i}) ^{-1}$. Let $A(G^φ)$ be adjacency matrix of $G^φ$. In this paper, we prove that $$m(G)-c(G)\leq p(G^φ)\leq m(G)+c(G),$$ $$m(G)-c(G)\leq n(G^φ)\leq m(G)+c(G),$$ where $p(G^φ)$, $n(G^φ)$, $m(G)$ and $c(G)$ are the number of positive eigenvalues of $A(G^φ)$, the number of negative eigenvalues of $A(G^φ)$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds, respectively.
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。
