of a graph $G$ is invariant under all $\mathbb{T}$-gains if and only if the $G$ is a tree\[Theorem \[spec-equi-tree\]\]. We identified a class of connected graphs such that for each graph $G$ in this class the cospectrality of matrices in $\mathbb{T}_G$ is determined by the gains of the fundamendal cycles.

Second objective of this article is to introduce a class of connected graphs $\mathfrak{F^{'}}$ such that for each $G$ in this collection, the matrices in $\mathbb{T}_G$ have nonnegative real part up to diagonal unitary similarity. We extend some of the bounds for the spectral radius of the $\mathbb{T}$-gain adjacency matrices of graphs in this class $\mathfrak{F^{'}}$, which are known for Hermitian adjacency matrices.

The spectral radius of the adjacency matrix $ A(G) $ of a graph $ G $ is the largest eigenvalue of $A(G)$. If we consider directed graphs or weighted graphs with complex numbers as edge weights, then the spectral radius need not be an eigenvalue of the associated adjacency matrix. For the Hermitian adjacency matrix of a digraph, surprisingly, in [@Bojan Theorem 5.6] the authors established the following bounds for the spectral radius of the Hermitian adjacency matrix $ H(X) $ in terms of the largest eigenvalue 