only if $ \Phi $ is balanced.

[@Our-gain1 Lemma 4.1] \[Theo.8\] Let $ \varphi $ be a $ \mathbb{T} $-gain on a connected graph $ G $, then $\rho(A(\Phi)) \leq \rho(A(G)) $.

[@Our-gain1 Theorem 4.4]\[Theo.9\] Let $ \varphi $ be a $ \mathbb{T} $-gain on a connected graph $ G $, then $\rho(A(\Phi))=\rho(A(G))$ if and only if either $\Phi$ or $ -\Phi $ is balanced.

The *characteristic polynomial* of an $n \times n$ matrix $ M $ is defined as $ det(M-xI) $, where $ I $ is the $n \times n$ identity matrix. A graph $G$ is called an *elementary graph*, if each of its component is either an edge or a cycle. Let $\mathcal{H}(G)$ denote the collection of all spanning elementary subgraphs of a graph $G$. For any $H \in\mathcal{H}(G)$, let $\mathcal{C}(H)$ denote the collection of all cycles in $H$.

\[Theo4\] Let $\Phi$ be any $\mathbb{T}$-gain graph with the underlying graph $G$. Let $P_\Phi(x)=x^n+a_1x^{n-1}+\cdots+a_n$ be the characteristic polynomial of $A(\Phi)$. Then $$a_i=\sum_{H\in\mathcal{H}_i(G)}(-1)^{p(H)}2^{c(H)}\prod_{C\in \mathcal{C}(H)}\Re(\varphi(C)),$$ where $\mathcal{H}_i(G)$ is the set of all elementary subgraphs of $G$ with $i$ vertices. $ p(H) $ and $ c(H) $ are the number of components and the number of cycles in $ H $, 