$\lambda_{1}(H(X))$ of $H(X)$.

[@Bojan Theorem 5.6]\[Theo.6\] For every digraph $X$, $ \lambda_{1}(H(X))\leq \rho(H(X))\leq3\lambda_{1}(H(X))$.

Third objective of this article is to extend the above result for $\mathbb{T}$-gain adjacency matrices. For any connected graph, we show that if the associated $\mathbb{T}$-gain graph has nonnegative real part, then the above bounds hold true\[Theorem \[Theo.5.1\]\]. Also, we have for any graph $G \in \mathfrak{F^{'}}$, the bounds in the above theorem hold true irrespective of the gains of the edges\[Theorem \[Theo.5.2\]\].

The spectral radius of the adjacency matrix of a graph is bounded above by the maximum vertex degree of the underlying graph. In [@Bojan], this bound was extended to Hermitian adjacency matrices. Also the sharpness of the bound has been characterized (See Theorem \[Theo.2.6\] of Section 2). Our fourth objective is to introduce the notion of $ k $-generalized Hermitian adjacency matrix, $ H_{k}(X) $ of a digraph $ X $, which extends the notion of Hermitian adjacency matrix of $X$. For this new class of matrices, we characterize the structure of a digraph $ X $ for which the spectral radius $ \rho(H_{k}(X))$ equals to the largest vertex degree of $\Gamma(X)$, which gives an alternative proof of .

Notation, definitions and known results
