if and only if $ Ax=\lambda_{max} x$ (respectively , $ Ax=\lambda_{min}x $). Moreover,

$ \lambda_{max}= \max_{x \neq 0}\frac{x^{*}Ax}{x^{*}x} $ and $ \lambda_{min}= \min_{x \neq 0}\frac{x^{*}Ax}{x^{*}x} $.

The *numerical range* of an $n \times n$ complex matrix $A$ is a subset of the complex numbers $ \mathbb{C} $, defined as follows:

$ W(A):=\{\langle Ax, x \rangle: x \in \mathbb{C}^n, \langle x, x \rangle=1  \} $.

For any $n \times n $ complex matrix $A$, the numerical range $W(A)$ is a convex set.

[@nrange] \[Theo.5\] If $A$ is an $n \times n$ Hermitian matrix, then $ W(A)$ is an interval $ [m,M] $, where $ m=\lambda_{min}(A), M=\lambda_{max}(A) $. Moreover, $ \rho(A) =   \max \{|m|,|M|\} $.

This article is organized as follows: In section 3, we study the cospectrality of matrices in $ \mathbb{T}_{G} $. First, we introduce an orientation scheme for the edges of the graph $G$, using a normal spanning tree of $G$, and we call the resultant oriented graph as suitably oriented graph. Suitably oriented graph plays an important role in this article. Using this notion, we construct uncountable many matrices in $ \mathbb{T}_{G} $ having the same spectrum. Then we give an equivalent condition for a graph $ G 