  Let $R$ be a Noetherian commutative ring of positive dimension. The
Hochster-Huneke graph of $R$ (sometimes called the dual graph of Spec $R$ and
denoted by $\mathcal{G} (R)$) is defined as follows: the vertices are the
minimal prime ideals of $R$, and the edges are the pairs of prime ideals
$(P_1,P_2)$ with height $(P_1 + P_2) = 1$. If $R$ satisfies Serre's property
$(S_2)$, then $\mathcal{G} (R)$ is connected. In this note, we provide lower
and upper bounds for the maximum diameter of Hochster-Huneke graphs of
Stanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number of
variables and the dimension. Hochster-Huneke graphs of $(S_2)$ Stanley-Reisner
rings are a natural abstraction of the $1$-skeletons of polyhedra. We discuss
how our bounds imply new Hirsch-type bounds on $1$-skeletons of polyhedra.
