as the origin of the graph $\mathbb{G} = (\mathcal{V},\mathcal{E})$. For $v \in \mathcal{V}$, we define $\vert v \vert$ as the length of the shortest path connecting $0$ to $v$. Hence, $\vert v \vert$ defines in the graph the distance from $0$ to $v$. For $r > 0$, let $S_r$ be the sphere of radius $r$ in the graph defined by $$S_r := \big\lbrace v \in \mathcal{V} : \vert v \vert = r \big\rbrace.$$

\[fig 1\]

![Spheres $S_r$ in the case of a binary tree graph, where $k = 2$.](tree_1.png)

In this case, we have $$\displaystyle \mathcal{V} = \bigsqcup_{0}^\infty S_r,$$ where $\bigsqcup$ means a disjoint union, so that we have $$\ell^2(\mathcal{V}) = \bigoplus_0^\infty \ell^2(S_r).$$ In this paper, we are interested in the case of regular rooted tree graphs with $k$-fold branching, $k \geq 1$. Moreover, the potential $M$ will be assumed to satisfy the following assumption:

[**Assumption (A)**]{}: For $ v \in \mathcal{V}$, we have $$\label{eq1,7}
\vert M(v)\vert \leq {\rm Const.} \, e^{-\delta \vert v \vert}, \quad {\rm with} \quad
\begin{cases} 
\delta > 0 & \text{if } \, k = 1, \\ 
\delta \geq 6 \ln (k) & {\rm otherwise}.
\end{cases}$$

\[r1,1\] We point out that in Assumption (A) above, there 