A $\mathbb{T}$-gain graph is a triple $\mathrm{\Phi }=(G,\mathbb{T},\varphi )$ consisting of an underlying graph $G=\left(V\right(G),E(G\left)\right)$, the circle group $\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}$ and a gain function $\varphi :\overrightarrow{E}\to \mathbb{T}$, such that $\varphi \left(e_{ij}\right)=\varphi (e_{ji}{)}^{-1}=\overline{\varphi \left(e_{ji}\right)}$. In this paper, we focus our attention on the relations between the inertia indices of $\mathbb{T}$-gain graph Φ and the inertia indices of its underlying graph G. We obtain sharp lower and upper bounds on $p\left(\mathrm{\Phi }\right)$ (resp., $n\left(\mathrm{\Phi }\right)$) in terms of $p\left(G\right)$ (resp., $n\left(G\right)$) and characterize those corresponding extremal $\mathbb{T}$-gain graphs, respectively. As a corollary, we also characterize those signed graphs whose inertia indices obtaining the sharp bounds, respectively.

一种$\mathbb{吨}$- 增益图是一个三元组$\mathrm{\Phi }=(G,\mathbb{吨},\phi )$由基础图组成$G=\left(五\right(G),乙(G\left)\right)$, 圆组$\mathbb{吨}=\{z\in \mathbb{C}：|z|=1\}$和增益函数$\phi ：\overrightarrow{乙}\to \mathbb{吨}$, 这样$\phi \left(e_{\mathrm{一世}j}\right)=\phi (e_{j\mathrm{一世}}{)}^{-1}=\overline{\phi \left(e_{j\mathrm{一世}}\right)}$. 在本文中，我们将注意力集中在惯性指数之间的关系上。$\mathbb{吨}$- 增益图 Φ 及其基础图G的惯性指数。我们在$p\left(\mathrm{\Phi }\right)$（分别，$n\left(\mathrm{\Phi }\right)$） 按照$p\left(G\right)$（分别，$n\left(G\right)$) 并表征那些相应的极值$\mathbb{吨}$-增益图，分别。作为推论，我们还分别描述了那些惯性指数获得锐界的有符号图。