Mixing Properties for Hom-Shifts and the Distance between Walks on Associated Graphs

Let $\mathcal{H}$ be a finite connected undirected graph and $\mathcal H_{walk}$bethegraphofbi-infinitewalkson$\mathcal H$;twosuchwalks${x_i}{i\in \mathbb Z}$and${y_i}{i \in \mathbb Z}$ are said to be adjacent if $x_i$ is adjacent to $y_i$ for all $i\in \mathbb{Z}$. We consider the question: Given a graph $\mathcal{H}$ when is the diameter (with respect to the graph metric) of ${\mathcal{H}}_{walk}$ finite? Such questions arise while studying mixing properties of hom-shifts (shift spaces which arise as the space of graph homomorphisms from the Cayley graph of ${\mathbb{Z}}^d$ with respect to the standard generators to $\mathcal{H}$) and are the subject of this paper.