Relations between the Positive Inertia Index of a $\mathbb{T}$-Gain Graph and That of Its Underlying Graph
Received:April 04, 2020  Revised:August 02, 2020
Key Word: complex unit gain graphs   inertia index
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11971474) and the Natural Science Foundation of Shandong Province (Grant No.ZR2019BA016).
 Author Name Affiliation Sai WANG School of Mathematics, China University of Mining and Technology, Jiangsu 221116, P. R. China Xuhai College, China University of Mining and Technology, Jiangsu 221116 P. R. China Dengyin WANG School of Mathematics, China University of Mining and Technology, Jiangsu 221116, P. R. China Fenglei TIAN School of Management, Qufu Normal University, Shandong 276826, P. R. China
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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.