Let R be an equidimensional commutative Noetherian ring of positive dimension. The dual graph of R is defined as follows: the vertices are the minimal prime ideals of R, and the edges are the pairs of prime ideals (P1, P2) with height (P1 + P2) = 1. If R satisfies Serre's property (S2), then G(R) is connected. In this note, we provide lower and upper bounds for the maximum diameter of dual graphs of Stanley-Reisner rings satisfying (S2). These bounds depend on the number of variables and the dimension. Dual graphs of (S2) 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.

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Throughout, let R be a commutative Noetherian ring. A ring R satisfies Serre's condition (S ℓ ) if for all p ∈ Spec R, depth Rp ≥ min{ℓ, dim Rp}. Serre's condition has been a topic of expanding interest. In this paper, we examine a generalization of Serre's condition (S j ℓ ). We say a ring satisfies (S j ℓ ) when depth Rp ≥ min{ℓ, dim Rp − j} for all p ∈ Spec R. We prove generalizations of results for rings satisfying Serre's condition.

We investigate generalized notions of the nerve complex for the facets of a simplicial complex. We show that the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring k[∆] as well as the f -vector and h-vector of ∆. We present, as an application, a formula for computing regularity of monomial ideals.

We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that if a balanced simplicial complex satisfies Serre's condition $(S_{\ell})$ then so do all of its rank selected subcomplexes.  We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial complex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi.