  Let $\mathcal H$ be a finite connected undirected graph and $\mathcal
H_{walk}$ be the graph of bi-infinite walks on $\mathcal H$; two such walks
$\{x_i\}_{i\in \mathbb Z}$ 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.
