takes integral values. For any $f:V\to {{\boldsymbol}{R}}$ we set for all $q\ge 1$, $U\subset V$ $$\begin{gathered}
  {{\lVertf\rVert}_{\ell^{q}(U)}}^{q}
  =
  \sum_{x\in U}
  {\lvertf(x)\rvert}^{q}
  {d_{{\omega}}}(x)
  \,,
  \quad
  {{\lVertf\rVert}_{\ell^{\infty}(U)}}
  =
  \sup_{x\in U} {\lvertf(x)\rvert}
  \,,
  \\
  {\mu_{{\omega}}}(U)
  =
  \sum_{x\in U}
  {d_{{\omega}}}(x)
  \,.\end{gathered}$$ All the infinite sums in this paper are absolutely convergent. In the following we always assume, unless explicitly noted, that all balls are centered at a given fixed $x_{0}\in V$ and we write $B_{R}(x_{0})=B_{R}$. We denote generic constants depending on the parameters of the problem by $\gamma$ (large constants), $\gamma_{0}$ (small constants). We also set for all $A\subset V$ $$\chi_{A}(x)
  =
  1
  \,,
  \quad
  x\in A
  \,;
  \qquad
  \chi_{A}(x)
  =
  0
  \,,
  \quad
  x\not\in A
  \,.$$

\[d:fk\] We say that $G$ satisfies a global Faber-Krahn inequality for a given $p>1$ and function ${\Lambda_p}:(0,+\infty)\to(0,+\infty)$ if for any $v>0$ and any finite subset $U\subset V$ with ${\mu_{{\omega}}}(U)=v$ we have $$\label{eq:fk}
    {\Lambda_p}(v)
    \sum_{x\in U}
    {\lvertf(x)\rvert}^{p}
    {d_{{\omega}}}(x)
    \le
    \sum_{x,y\in (U)_{1}}
    {\lvertf(y)-f(x)\rvert}^{p}
    {\omega}(x,y)
    \,,$$ for all $f:V\to {{\boldsymbol}{R}}$ such that $f(x)=0$ if $x\not \in U$; here $$(U)_{1}
    =
    \{
    x\in V
    \mid
    d(x,U)
    \le 1
    \}
    \,.$$

We assume throughout that ${\Lambda_p}\in C(0,+\infty)$ is decreasing and that two suitable positive constants ${N}$, $\omega$ exists such that $$\begin{aligned}
  \label{eq:fkf_nd}
  