Let
$\mathcal{\mathscr{H}}$ be a finite connected
undirected graph and
${\mathcal{\mathscr{H}}}_{\mathrm{\text{walk}}}^{2}$ be the
graph of bi-infinite walks on
$\mathcal{\mathscr{H}}$;
two such walks
${\left\{{x}_{i}\right\}}_{i\in \mathbb{Z}}$
and
${\left\{{y}_{i}\right\}}_{i\in \mathbb{Z}}$ 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
$\mathcal{\mathscr{H}}$,
when is the diameter (with respect to the graph metric) of
${\mathcal{\mathscr{H}}}_{\mathrm{\text{walk}}}^{2}$
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
$\mathcal{\mathscr{H}}$)
and are the subject of this paper.