$ to be a tree in terms of the spectrum and the spectral radius of matrices in $ \mathbb{T}_{G} $. Next, we give a class of connected graphs such that for each graph $ G $ in this class, the cospectrality of matrices in $ \mathbb{T}_{G} $ is characterized in terms of the real part of gains of their cycles.

In section 4, we introduce a new class of connected graphs, $ \mathfrak{F^{'}} $ in terms of $ \mathbb{T}$-gains. More precisely, if $ G \in \mathfrak{F^{'}}$, then the matrices in $ \mathbb{T}_{G} $ have nonnegative real part up to diagonal unitary similarity. In section 5, we discuss bounds for the spectral radius of matrices in $ \mathbb{T}_{G} $. First, we obtain bounds of the spectral radius in terms of the largest eigenvalue of matrices in $ \mathbb{T}_{G} $, where $ G \in \mathfrak{F^{'}} $. Next, we establish a characterization for the gain graph $ \Phi = (G, \varphi) $ to have $ \rho(A(\Phi)) =\Delta $. Then, for any digraph $ X $, we define the notion of $k$-generalized Hermitian adjacency matrix, $ H_{k}(X) $, which is an extension of the notion of Hermitian adjacency matrix. Finally, we characterize digraph $ 