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Let $R$ be a Noetherian commutative ring of positive dimension. TheHochster-Huneke graph of $R$ (sometimes called the dual graph of Spec $R$ anddenoted by $\mathcal{G} (R)$) is defined as follows: the vertices are theminimal 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 lowerand upper bounds for the maximum diameter of Hochster-Huneke graphs ofStanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number ofvariables and the dimension. Hochster-Huneke graphs of $(S_2)$ Stanley-Reisnerrings are a natural abstraction of the $1$-skeletons of polyhedra. We discusshow our bounds imply new Hirsch-type bounds on $1$-skeletons of polyhedra.

On the diameter of dual graphs of Stanley-Reisner rings with Serre $(S_2)$ property and Hirsch type bounds on abstractions of polytopes