增益图与其底图的正惯性指数之间的关系

设$\mathbb{T}$是模为1的复数乘法子群.图$G=(V,E)$,这里$V,E$分别表示图的点和边.增益图是将底图中的每条边赋于$\mathbb{T}$中的某个数值$\varphi(v_iv_j)$,且满足$\varphi(v_iv_j) =\overline{\varphi(v_jv_i)}$.将赋值以后的增益图表示为$(G,\varphi)$.设$i_+(G,\varphi)$和$i_+(G)$分别表示增益图与底图的正惯性指数,本文证明了如下结论: $$ - c( G ) \le {i_ + } ( {G,\varphi } ) - {i_ + }( G ) \le c( G ), $$ 这里$c(G)$表示圈空间维数,并且刻画了等号成立时候的所有极图.

Relations between the Positive Inertia Index of a $\mathbb{T}$-Gain Graph and That of Its Underlying Graph

Let $\mathbb{T}$ be the subgroup of the multiplicative group $\mathbb{C}^\times$ consisting of all complex numbers $z$ with $| z | = 1$. A $\mathbb{T}$-gain graph is a triple $\Phi =(G,\mathbb{T}, \varphi)$ ( or short for $(G,\varphi)$ ) consisting of a simple graph $G = (V,E)$, as the underlying graph of $(G,\varphi)$, the circle group $\mathbb{T}$ and a gain function $\varphi:\overrightarrow{E} \to \mathbb{T}$ such that $\varphi(v_iv_j) = \overline{\varphi(v_jv_i)}$ for any adjacent vertices $v_i$ and $v_j$. Let $i_+(G,\varphi)$ (resp., $i_+(G)$ ) be the positive inertia index of $(G,\varphi)$ (resp., $G$). In this paper, we prove that $$ - c( G ) \le {i_ + } ( {G,\varphi } ) - {i_ + }( G ) \le c( G ), $$ where $c(G)$ is the cyclomatic number of $G$, and characterize all the corresponding extremal graphs.