Abstract
We investigate the average range of 1-Lipschitz mappings (graph-indexed random walks) of a given connected graph. This parameter originated in statistical physics, it is connected to the study of random graph homomorphisms and generalizes standard random walks on Z.
Our first goal is to prove a closed-form formula for this parameter for cycle graphs. The second one is to prove two conjectures, the first by Benjamini, Häggström and Mossel and the second by Loebl, Nešetřil and Reed, for unicyclic graphs.
This extends a result of Wu, Xu, and Zhu [5] who proved the aforementioned conjectures for trees.