also considered in several papers dealing with features like blow up or extinction; we quote only [@Chung:Choi:2014] and [@Chung:Park:2017].

The case of the discrete linear Laplacian where $p=2$ is more classical, also for its connections with probability theory (see e.g., [@Andres:etal:2013] and references therein), and is often attacked by means of suitable parallels with the theory of heat kernels in manifolds. We quote [@Coulhon:Grigoryan:1998], [@Barlow:Coulhon:Grigoryan:2001] where a connection is drawn between properties of heat kernels, of graphs and Faber-Krahn functions.\
In [@Lin:Wu:2017] heat kernels are used to study the blow up of solutions to the Cauchy problem for a semilinear equation on a possibly infinite graph.

The subject of diffusion in graphs is popular also owing to its applicative interest. We refer the reader to [@Mugnolo:2013], [@Elmoataz:Toutain:Tenbrinck:2015] and to the references therein for more on this point.

Finally we recall the papers [@Bakry:etal:1995a], [@Ostrovskii:2005] and books [@Chung:SGT], [@Grigoryan:AG] for basic information on functional analysis on graphs and manifolds.

We mention that in our setting it is still valid the argument in [@Bonforte:Grillo:2007] showing that optimal decay rates imply suitable embeddings.

Here we look essentially at the approach of [@DiBenedetto:Herrero:1989] and [@Andreucci:Tedeev:2015].

The paper is organized as follows: Section \[s:prelim\] is 