new, as far as we know.

\[t:l1\] Let ${u}_{0}\in\ell^{1}(V)$, ${u}_{0}\ge 0$. Then problem – has a unique solution satisfying for all $t>0$ $$\begin{aligned}
    \label{eq:l1_n}
    {{\lVert{u}(t)\rVert}_{\ell^{1}(V)}}
    &=
      {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}
      \,,
    \\
    \label{eq:l1_nn}
    {{\lVert{u}(t)\rVert}_{\ell^{\infty}(V)}}
    &\le
      \gamma
      {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}
      {\psi}_{1}^{(-1)}
      \big(
      t^{-1}
      {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}^{-(p-2)}
      \big)
      \,.
  \end{aligned}$$ In addition ${u}$ satisfies $$\begin{gathered}
    \label{eq:l1_nnn}
    \int_{0}^{t}
    \sum_{x,y\in V}
    {\lvert{D}_{y}{u}(x,\tau)\rvert}^{p-1}
    {\omega}(x,y)
    {\,\textup{\textmd{d}}}\tau
    \\
    \le
    \gamma
    t^{\frac{1}{p}}
    {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}^{\frac{2(p-1)}{p}}
    {\psi}_{1}^{(-1)}(t^{-1}{{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}^{-(p-2)})^{\frac{p-2}{p}}
    \,.
  \end{gathered}$$

\[r:lp\_est\] We notice that one could exploit , to derive trivially a bound of the integral in . This is due of course to the fact that the $p$-laplacian in our setting is discrete, and it would not be possible in the framework of continuous partial differential equations.

Such a bound however is not sharp, and for example could not be used in the proof of Theorem \[p:bbl\].

In other instances where optimality is not needed we exploit a device similar to the one just described, relying on Remark \[r:lp\_scale\]; see for example the proof of Lemma \[l:cacc2\].

So far our extension to graphs of methods and results of continuous differential equations has been successful. However, in the latter setting a standard device to prove optimality of the bound in relies on the property of finite speed of propagation 