where $M$ is identified with the multiplication operator by the bounded potential function (also) noted $M$. In a regular rooted tree graph with $k$-fold branching, according to above, the operator ${{-\Delta}}_M$ can be written as $$\label{eq1,6}
{{-\Delta}}_M = -L + k + 1 - d_0 + M.$$ In , the degree term $d_0$ can be included in the potential perturbation so that ${{-\Delta}}_M$ can be viewed as a perturbation of the operator $-L + k + 1$. Hence, from now on, the operator ${{-\Delta}}_M$ will be written as $$\label{eq1,60}
{{-\Delta}}_{\widetilde M} = -L + k + 1 + \widetilde M \quad {\rm with} \quad 
\widetilde M := - d_0 + M.$$ In the sequel, we set $$\label{eq1,61}
t_\pm(k) := \pm 2\sqrt{k} + k + 1,$$ and we shall simply write $t_\pm$ when no confusion can arise. Then, from , it follows that the spectrum of the operator $-L + k + 1$ satisfies $$\label{eq1,7}
\sigma (-L + k + 1) = \sigma_{\textup{ac}} (-L + k + 1) = \sigma_{\textup{ess}} (-L + k + 1) = 
\left[ t_-(k),t_+(k) \right],$$ where the $t_\pm(k)$ play the role of thresholds of this spectrum.

Now, let us choose some vertex $v_0 = 0 \in \mathcal{V}$ 