\sim v} \phi(w),$$ then it is not difficult to see that the operator ${{-\Delta}}$ can be written as $$\label{eq1,3}
{{-\Delta}}= -L + d,$$ where $d$ is the multiplication operator by the function (also) noted $d : \mathcal{V} \rightarrow {{\mathbb{C}}}$, with $d(v)$ denoting the number of edges connected with the vertex $v$. Note that when $d$ is bounded, then so is the symmetric operators ${{-\Delta}}$ and $L$, hence self-adjoint. In a regular rooted tree graph with $k$-fold branching, $k \ge 1$, (see Figure 2.1 for a binary tree graph), we have $d = k + 1 - d_0$ with $$d_0(v) = \begin{cases} 
1 & \text{if } v \text{ coincides with the root of the tree}, \\ 
0 & \text{otherwise}. 
\end{cases}$$ This is the same model described in [@af] and we refer to this paper for more details. In , $d$ can be viewed as a perturbation of the operator $L$. It is well know (see Lemma \[l3,2\]) that the spectrum of the operator $L$ is absolutely continuous, coincides with the essential spectrum and is equal to $$\label{eq1,4}
\sigma (L) = \sigma_{\textup{ac}} (L) = \sigma_{\textup{ess}} (L) = 
\left[ -2\sqrt{k},2\sqrt{k} \right].$$

On $\ell^2(\mathcal{V})$, we define the perturbed operator $$\label{eq1,5}
{{-\Delta}}_M := {{-\Delta}}+ M,$$ 