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

Abstract:

      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.