A bounded linear operator $T$ on a Banach space $X$ is called power bounded if $\|T^k\|\le M$ for some $M>0$ and all $k\in \mathbb N$.
A classical result of Lorch says that if $X$ is reflexive, then each power bounded operator $T$ on it is mean ergodic, meaning that the sequence $$ C_n x:= \frac{1}{n} \sum_{k=1}^n T^k x,\qquad n=1,2,\ldots, $$ of its Cesaro sums converges for all $x\in X$. (Actually, also the converse implication holds).
Now, take a simple digraph (possibly infinite, but uniformly locally finite) and consider its adjacency matrix $A$. It would be nice to know whether $(A^n)_{n\in \mathbb N}$ converges, but this seems to be usually hopeless. Hence an more pragmatic approach would be to investigate whether $A$ is at least mean ergodic. By the above result by Lorch, it would be sufficient to show power boundedness.
Hence my question:
Is it possibly known whether $A$ is power bounded for certain classes of digraphs?
EDIT: As Robert Israel points out in his answer, the question is trivial if the graph is not oriented. Therefore, I have restricted the question to digraphs.
