v&\mapsto
     {\Lambda_p}(v)^{-1}
     v^{-\frac{p}{{N}}}
     \,,
     \quad
     v>0
     \,,
     \quad
     \text{is nondecreasing;}
  \\
  \label{eq:fkf_ni}
  v&\mapsto
     {\Lambda_p}(v)^{-1}
     v^{-\omega}
     \,,
     \quad
     v>0
     \,,
     \quad
     \text{is nonincreasing.}\end{aligned}$$ An important class of functions in our approach is given by $$\label{eq:dcf_def}
  {\psi}_{r}(s)
  =
  s^{\frac{p-2}{r}}
  {\Lambda_p}(s^{-1})
  \,,
  \qquad
  s>0
  \,,$$ for each fixed $r\ge 1$. They, or more exactly their inverses, give the correct time-space scaling for the equation , see for example Theorems \[t:l1\] and \[p:bbl\] below.

If we make the additional assumption that for some constant $c>0$ $$\label{eq:fkf_bbl}
  {\Lambda_p}(v)
  \ge
  c
  {\mathcal{R}}(v)^{-p}
  \,,
  \qquad
  v>0
  \,,$$ where ${\mathcal{R}}:(0,+\infty)\to (0,+\infty)$ is such that ${\mu_{{\omega}}}(B_{{\mathcal{R}}(v)})=v$, we may connect ${\psi}_{1}$ to the measure of a ball in $G$. This in turn allows us to prove sharpness of our $\ell^{1}$–$\ell^{\infty}$ estimate. Property is rather natural. For instance it is known to hold for the explicit examples in Subsection \[s:examples\], to which we refer for implementations of our results in some concrete relevant cases.

\[r:spdim\] The constant ${N}$ in has no intrinsic meaning in this paper, and it is employed here only with the purpose of making easier the comparison with the case of standard regular graphs ${{\boldsymbol}{Z}}^{{N}}$, where ${\Lambda_p}(v)=\gamma_{0}v^{-p/{N}}$, see Subsection \[s:examples\].

\[r:fk\_below\] Let $x\in V$ and choose $U=\{x\}$, $f=\chi_{U}$ 