Liouville property of graphs

Remarks

Example: $\Z^d$ with unit weights has the Liouville property for every $d \ge 1$. For $d = 1$ this is direct, since the harmonic functions on $\Z$ are $f(n) = an + b$, and such a function is bounded only when $a = 0$. Non-example: the $3$-regular tree $T_3$ fails it. Fix an edge $\set{o, o'}$, and let $h(x)$ be the probability that the simple random walk from $x$ eventually stays in the component of $T_3$ minus this edge that contains $o'$. The walk on $T_3$ is transient, so it crosses the edge finitely often, and $h$ is harmonic by conditioning on the first step. Its values lie in $[0, 1]$, and it is not constant. Indeed $h(x)$ tends to $1$ as $x$ moves away from the edge on the side of $o'$, and to $0$ on the side of $o$. The strong Liouville property implies the Liouville property, since a bounded harmonic $f$ gives the positive harmonic function $f - \inf f + 1$.