Let \(R\) be an equidimensional commutative Noetherian ring of positive dimension. One says that \(R\) satisfies the Serre’s condition (\(S_{\ell}\)), if for every \(P\in \mathrm{Spec}(R)\), the inequality \(\text{depth}\;R_P\geq \min\{\ell,\dim R_P\}\) holds true. The dual graph \(\mathcal{G}(R)\) of \(R\) 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 \(\text{height}(P_1+P_2)=1\). It is well-known that \(\mathcal{G}(R)\) is a connected graph provided that \(R\) satisfies the Serre’s condition (\(S_2\)). Let \(\mu(d,n)\) be the largest diameter of a dual graph of an (\(S_2\)) Stanley-Reisner ring of dimension \(d\) and codimension \(n-d\). In the paper under review, the author provides lower and upper bounds for \(\mu(d,n)\). For example, it is shown that
(i) \(\mu(3,n)\leq \max(2n-10, n-2)\), for any \(n\geq 3\),
(ii) \(\mu(d,n) \leq 2^{d-2}(n-d)\), for all \(d\geq 2\) and all \(n\geq d\),
(iii) \(\mu(d,d+k)\leq 3.2^{\frac{n-d-5}{2}}(n-d)\), for all \(d\geq 2\),
(iv) \(\mu(4,4k+4)\geq 6k\),
(v) \(\mu(3,8k+2) \geq 10k-1\) and
(vi) \(\mu (3,8k+3+j) \geq 10k+j+1\), for all \(j\geq 4\).
The bounds are used to determine \(\mu(d,n)\) for small values of \(d\) and \(n\).
Dual graphs of (\(S_2\)) Stanley-Reisner rings are a natural abstraction of the \(1\)-skeletons of polyhedra. The author discusses how his results imply new Hirsch-type bounds on \(1\)-skeletons of polyhedra.