. Is there a master theorem that can be used to determine whether or not a simple random walk (choose a random neighboring vertex as the next step) on a given infinite graph leads to recurrence (almost surely returns to the start vertex)?Q

**Added**. Let's restrict the graphs to be locally finite, in the sense
that every vertex has finite degree.

For example, the infinite path is recurrent, as it is equivalent to $\mathbb{Z}^1$, but I believe the infinite binary tree is not recurrent, i.e., it is transient. I am seeking structural properties of a graph that determines its recurrence/transience behavior.