On the Diameter of Dual Graphs of Stanley-Reisner Rings and Hirsch Type Bounds on Abstractions of Polytopes

Let R be an equidimensional commutative Noetherian ring of positive dimension. The dual graph 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 (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.


Introduction
The polynomial Hirsch conjecture states that a d-dimensional polyhedron with n facets has diameter bounded above by a polynomial expression in n − d.The diameter of a polyhedron is the diameter of its 1-skeleton.The polynomial Hirsch conjecture is a weakening of the Hirsch conjecture.The Hirsch conjecture was disproved by Klee and Walkup [17] in the general case and Santos [22] in the bounded case.For a history of the Hirsch and polynomial Hirsch conjectures, see [23].In this paper we construct bounds which improve on bounds from existing literature [18,2,8], but are not polynomial.Our bounds are sharp for small n and d.Many authors have examined the diameters of generalizations of polyhedra (e.g.[1,6,8,16]).We consider generalizations of polyhedra the electronic journal of combinatorics 25(1) (2018), #P1.60 whose 1-skeletons G have vertices that are subsets of size d of {1, 2, . . ., n}, such that G has the following properties (see Section 1 of [8]): 1.For each u, v ∈ V (G) there exists a path connecting u and v whose intermediate vertices all contain u ∩ v.
2. The edge (u, v) is present if and only if |u ∩ v| = d − 1.
Condition (2) is equivalent to G is the adjacency graph of a pure simplicial complex and (1) says that simplicial complex is "normal" as defined in Definition 3.1 in [23].Generalized polyhedra of this type have been considered in section 4.1 of [16].
Dual graphs are an object of wide interest in commutative algebra and algebraic geometry (e.g.[12,3,4,5,20]).From our setting, we shall consider the dual graph to have vertices corresponding to the minimal primes of a ring, however, the dual graph can be constructed in a more general setting with vertices corresponding to the irreducible components of a scheme.It is a famous result of Hartshorne [12] that if X is a connected projective scheme such that O X,x satisfies Serre's condition (S 2 ) for all x ∈ X, then the dual graph of X is connected.This result is commonly known in its less general form to say arithmetically Cohen-Macaulay projective schemes have connected dual graphs.
Stanley-Reisner rings satisfying (S 2 ) have recently attracted much attention [19,10,21,7].It is known that a graph having properties (i) and (ii) is equivalent to that graph being the dual graph of a Stanley-Reisner ring satisfying Serre's condition (S 2 ).We shall combine techniques from commutative algebra and combinatorics to prove bounds on the diameter of these graphs.
We define µ(d, n) to be the largest diameter of a dual graph of an (S 2 ) Stanley-Reisner ring of dimension d and codimension n − d.One of the main results of this paper is the determination of the precise values of µ(d, n) for small n and d (see Table 1).
To that end we first construct upper bounds for quite general n and d.For instance, Theorem 19 shows µ (3, n) max(2n − 10, n − 2).In Theorem 22, we prove µ(d, n) 2 d−2 (n − d), which improves on the bound of [8] (See Remark 23).In Theorem 26, we prove This result is derived using our bound from Theorem 22. Combining these results with manually generated constructions (Appendix A), we can produce the following table of exact values of µ(d, n): In Section 6 of [22], Santos builds arbitrarily large complexes whose diameters exceed the Hirsch bound by a fixed fraction.This is achieved by using a gluing lemma from [15], the electronic journal of combinatorics 25(1) (2018), #P1.60 which states that two d-dimensional polytopes, P 1 , P 2 , can be glued together yielding a new polytope P with diam P diam P 1 + diam P 2 − 1.We present Theorem 28, an algebraic analogue to the gluing lemma.This theorem tells us that two d − 1-dimensional complexes with (S ) rings (we shall call these (S ) complexes) glued together along a pure, (S −1 ) subcomplex of dimension at least d − 2 yield an (S ) complex.The proof of Theorem 28 is achieved using local cohomology and local duality.
Applying Theorem 28, we are able to construct complexes whose Stanley-Reisner rings have dual graphs with arbitrarily large diameter which (with proper labeling) have properties (i) and (ii).For appropriate complexes ∆ and ∆ , diam Gluing multiple copies of examples from the small n and d cases together, we construct graphs with properties (i) and (ii) in dimensions 3 and 4 with diameters 5  4 (n − d) and  A polyhedron whose vertices are the intersection of d of its facets is non-degenerate.Any polyhedron can be transformed into a non-degenerate polyhedron by perturbation without decreasing its diameter [8].Therefore, we may restrict our attention to nondegenerate polyhedra.Definition 4. A pure simplicial complex is a simplicial complex whose facets all have the same dimension.
We remind the reader that a (d−1)-dimensional simplicial complex has a d-dimensional Stanley-Reisner ring.We will use as notation ∆ R to be the simplicial complex with Stanley-Reisner ring R. We shall use ∆ when the ring is either unspecified or clear from context.
Let k be a field and S = k[x 1 , . . ., x n ].Let ∆ be a pure, (d − 1)-dimensional simplicial complex with Stanley-Reisner ring R = S/I, where I is the intersection of the minimal prime ideals P i of R. Let {F i } be the facets of ∆.With proper numbering, P i is generated by {x j |x j / ∈ F i } (see e.g. the survey [9] for proof).Thus purity of ∆ is equivalent to each P i being generated by n − d distinct variables.
Let S/I be the Stanley-Reisner ring of ∆, S/I ∨ be the Stanley-Reisner ring of ∆ ∨ .We refer to I ∨ as the Alexander dual of I.The Alexander dual of I is generated by the product of the generators of each minimal prime ideal of I (see e.g. the survey [9] for proof).
Remark 8. We notice that the vertices of G(S/I) are in one to one correspondence with the generators of I ∨ .Also, each vertex is comprised of n − d variables.Finally, notice that by definition every dual graph has property (ii).Definition 9. Given a graph G(R), we define Ḡ(R), to be the graph with The graph Ḡ(R) is a relabeling of G(R).We construct this labeling so that our graphs fit the setting of [8], and so that we can determine if our graphs have properties (i) and (ii).
Remark 10.We note Ḡ(R) is the facet-ridge graph of the complex with Stanley-Reisner ring R. Definition 11.A graph Ḡ(R) is locally connected if for any two vertices vi , vj ∈ Ḡ(R), there exists a path between them such that each vertex in the path contains vi ∩ vj .
Locally connected is also referred to as ultra connected in [16].
Remark 12. Locally connected graphs are commonly referred to as normal graphs in combinatorial circles.Normality of a graph is a notion unrelated to normality of a ring.To avoid this confusion, we use the name "locally connected," which has been motivated by Theorem 16, which connects this property of Ḡ(R) to localization of the ring R.

Serre's condition
We have shown graphs with properties (i) and (ii) from [8] are an abstraction of 1skeletons of non-degenerate polyhedra.Ensuring a graph satisfies property (ii) is not difficult.Using Serre's condition and syzygy matrices, we demonstrate a simple method to ensure a dual graph of an (S 2 ) Stanley-Reisner ring satisfies property (i).
Definition 13.A ring satisfies Serre's condition (S ) if for all P in Spec R, A first syzygy matrix of a module is a matrix whose columns span all the first syzygies of that module.
Theorem 15 (Yanagawa).The Stanley-Reisner ring R = S/I satisfies (S 2 ) if and only if R is equidimensional and I ∨ has a first syzygy matrix with only linear entries.
This theorem is true by Corollary 3.7 in [28].Since checking the linearity of a first syzygy matrix can be easily done computationally, this theorem provides us with a simple way to demonstrate if a Stanley-Reisner ring satisfies (S 2 ).Now we connect (S 2 ) to our locally connected condition.We will use local duality, the ext module, and the cohomology module.For background on these topics, see a Homological Algebra Text (e.g.[27]).
Theorem 16.Let R be an equidimensional Stanley-Reisner ring.The following are equivalent: 2. For any prime ideal P generated by variables, G(R P ) is connected.

Ḡ(R) is locally connected.
the electronic journal of combinatorics 25(1) (2018), #P1.60 connected for any choice of S. By the definition of Ḡ(R), Ḡ(R) S is a relabeling of a subgraph of G(R) whose vertices are the minimal prime ideals of R contained in P , the prime generated by {x j |x j / ∈ S}.This subgraph is G(R P ).Therefore G(R P ) connected for all primes generated by variables is equivalent to Ḡ(R) being locally connected.
(1) ⇒ (2): Suppose G(R P ) is not connected for some nonempty set of primes generated by variables in Spec R. Note that every prime in this set must have height at least 2. Let us choose any prime P maximal in this set.If G(R P ) is not connected then Ḡ(R P ) is not connected.Suppose Ḡ(R P ) contains two distinct components each of which contain x i ∈ Supp R P .Then Ḡ(R P ) can be localized at S = x i yielding a disconnected graph, contradicting the maximality of P .Therefore, each connected component of Ḡ(R P ) is composed of disjoint sets of variables.The vertices of Ḡ(R P ), however, represent the facets of the simplicial complex ∆ R P .Therefore, we have that ∆ R P is not connected, which implies H 1 P R P (R P ) = 0. Thus depth R P 1; however, ht P 2, and thus dim R P 2. Therefore, R is not (S 2 ).
(1) ⇐ (2): Using local duality, we have a ring R satisfies (S 2 ) if and only if dim Ext n−i S (R, ω S ) i − 2 for all i < d (see Lemma 27).From [28], since R is a Stanley-Reisner ring, Ext n−i S (R, ω S ) is a square-free module.Thus Ext n−i S (R, ω S ) is uniquely determined by its primes generated by variables.The dimension of Ext n−i S (R, ω S ) determines if R satisfies (S 2 ).Therefore, we only need consider primes generated by variables when showing that R satisfies (S 2 ).
If G(R P ) is connected for all primes generated by variables in Spec R, then ∆ R P is connected for all primes generated by variables with height at least 2 in Spec R. When ht P 2, ∆ R P is connected if and only if H 1 P R P (R P ) = 0. Further, H 0 P R P (R P ) = 0 for all P ∈ Spec R, since R is a Stanley-Reisner ring.Thus depth R P 2 for all primes generated by variables with height at least 2, and depth R P 1 for all primes generated by variables with height 1.Thus R satisfies (S 2 ).Definition 17.Given a graph G with each vertex labeled with the same number of variables the following are equivalent: 1. G has properties (i) and (ii) from [8].Proof.
(1) ⇒ (2): Let each vertex label of G be a facet of a pure simplicial complex ∆.Let the Stanley-Reisner ring of ∆ be R. Then G has the same vertex set as Ḡ(R).Property (ii) implies that G has the same edge set as the dual graph of R. Property (i) is equivalent to locally connected.
(1) ⇐ (2): Property (ii) is required by the definition of a dual graph.Property (i) is precisely the same as local connectedness.

Upper Bounds
In this section we prove upper bounds for the diameters of dual graphs of (S 2 ) Stanley-Reisner rings.This is achieved by working with Ḡ(R) (see Section 2.1).
Definition 18.A strictly increasing path is a path such that each subsequent vertex in the path has larger distance from the starting vertex of the path.
Proof.We construct this upper bound in a manner inspired by [8].Let R be an (S 2 ) Stanley-Reisner ring, such that Ḡ(R) has vertices ABC, DEF with maximum distance in the graph.Let us assign to each vertex v the integer dist(ABC, v), where dist(ABC, v) denotes the length of the shortest path from ABC to v. We define layers Next, we construct a second block.Let d be the largest integer such that L d contains a variable of L c ; call this variable a 1 .Let B 2 = {L i |c < i d}.Let n 1 be the number of variables in B 2 but not in B 1 .The diameter of this layer will be bounded by the maximum length of a path in which each vertex contains a 1 .This path will have maximum length of n 1 + n 0 − 3 − 3 (the second −3 is to account for the fact that A, B, C cannot be in the layers of this block).
Construct the third block B 3 in the same way.Its longest path will have maximal length n 2 + n 1 − 3 − 3.By construction, B 3 cannot have any elements in common with B 1 .Also, B 3 does not contain any variables in the c + 1 layer (there are at least 3 such variables).
Continue in this manner.
We sum the lengths of the blocks and add 1 for each path between blocks to obtain: where k is the number of blocks.
the electronic journal of combinatorics 25(1) (2018), #P1.60 If ADE, ADF , or AEF is a vertex, then our path containing A has length at most n − 3, and some vertex in that path is adjacent to DEF .Therefore, diam Ḡ(R) n − 2.
Let L j be the largest layer containing the vertex x i .Define Denote the vertex containing AD in L i * to be v * (if more than one exists choose any such one).
Consider the case L A = n − 3. To maximize i * , we must have a path with tail: Proof.Let R be a codimension-4, (S 2 ) Stanley-Reisner ring of dimension at least 4. Take v 1 , v 2 ∈ V ( Ḡ(R)).Then, v 1 , v 2 will each contain n − 4 variables, and v 1 ∩ v 2 will contain at least n − 8 variables.Thus there must be a path from v 1 to v 2 in which each vertex in that path contains those n − 8 shared variables.Thus µ(d, d + 4) µ(4, 8).Furthermore, we may take the graph in Figure 8  Proof.Choose any v 1 , v 2 ∈ Ḡ(R).As before, we can reduce to the d 3, then diam Ḡ(R) is bounded by 7 (the d = 3 n = 9 case, see Table 1).Now suppose degree(v 1 ∩ v 2 ) 2. There must exist a vertex v 3 such that degree(v 3 ∩ v 1 ) 3 and degree(v 3 ∩ v 2 ) 3.
Theorem 26.For d 2: Proof.We only need to consider the case (n − d, 2(n − d)).
We will first consider the case n − d is even.In this case, Next we consider the case n − d is odd.Then

Gluing
In [22], Santos uses a gluing lemma to construct polyhedra with arbitrarily many facets whose diameters exceed n − d by a fixed fraction.We will construct an algebraic analogue to this gluing lemma, which will allow us to construct (S ) complexes with arbitrarily many facets whose diameters exceed n − d by a fixed fraction.The facet-ridge graph of an (S ) complex is Ḡ(R), where R = S/I is that complex's Stanley-Reisner ring.Thus by making these (S 2 ) complexes, we are making dual graphs of Stanley-Reisner rings satisfying (S 2 ) with arbitrarily large n whose diameters exceed n − d by a fixed fraction.
In this section, we will make use of the ext module Ext i S (R, S) and the cohomology module H i P R P (R P ).For background on these modules, see a text on Homological Algebra (e.g.[27]).We can describe (S ) in terms of the dimension of the Ext module.This fact is key to the proof that proper gluing will maintain (S ).The following lemma appears without proof in [26,Proposition 3.1].
Proof.Start with the graph made of k copies of Figure 8. Then append the vertex For the d = 3 case, we will consider three graphs: The graph G 0 from Figure 6, the graph G 1 from Figure 7, and the graph G 2 , which is the graph G 1 with {I, J, K} appended to the end.Proof.Take the graph constructed in Theorem 31.Append the vertices {n − 2 + i, n − 1 + i, n+i} for i = 1, . . ., j +1.This will be a locally connected graph of diameter 10k +j.

Final Remarks
These results raise several questions for further study.Primarily, is µ(d, n) bounded above by a polynomial in the codimension?A positive answer to this question would affirm the polynomial Hirsch conjecture.Our work still leaves the possibility that µ(d, n) is bounded by a linear function.
Another interesting question is, do the bounds of Larman [18] and Barnette [2] hold for larger values of d?We have seen that the bounds of Larman and Barnette do not hold for our d = 3 case.Figure 1  we can cone over this complex to show the bound of Barnette does not hold for d = 5; however, we do not have counterexamples in any higher dimensions.
In this paper we found graphs of maximal diameter for (S 2 ) Stanley-Reisner rings with d = 3, 4 and small n.We then used those graphs in conjunction with gluing to make graphs with large diameters with respect to codimension.It would be valuable to know what the largest diameter would be for graphs of (S 2 ) Stanley-Reisner rings with d = 5, 6 and small n, specifically, d = 5, n = 10 and d = 6, n = 12.Answers to these questions could lead to new asymptotic lower bounds and could give insight on how these bounds would grow with respect to d.

A Constructing graphs to identify lower bounds
In this appendix, for d and n fixed, we construct lower bounds for the maximum diameters of dual graphs of equidimensional Stanley-Reisner rings satisfying (S 2 ).We achieve this by constructing graphs with properties (i) and (ii).Recall, µ(d, n) is the largest diameter of a dual graph of an equidimensional, (S  Proof.R satisfies (S 2 ) if and only if Ḡ(R) is locally connected.In the d = 2 case, locally connected is equivalent to connected.Thus we construct a connected graph.Create a vertex v 1 = x 1 x 2 .We wish to create another vertex v 2 adjacent to v 1 .Without loss of generality, v 2 = x 1 x 3 .Any vertex not adjacent to v 1 but adjacent to v 2 must be of the form x 3 x i (i = 1, 2, 3).Thus we may choose v 3 = x 3 x 4 .Continuing this process, we see that µ(2, n) is bounded above by the number of variables in R that are not contained in v 1 .We also see that this construction yields a graph of diameter n − 2. Thus µ(2, n) = n − 2. Figure 3 shows a graph with properties (i) and (ii) of diameter 3 when n = 5, d = 2. Proof.Let us consider any distinct pair of vertices Proof.Let R be a codimension-3, (S 2 ) Stanley-Reisner ring.Take v 1 , v 2 ∈ V ( Ḡ(R)).Then, v 1 , v 2 will each contain n − 3 variables, and v 1 ∩ v 2 will contain at least n − 6 variables.Thus there must be a path from v 1 to v 2 in which each vertex in that path contains those n − 6 shared variables.Thus µ(d, d + 3) µ (3,6).Furthermore, we may take the graph with vertex set {ABC, BCD, CDE, DEF } and add the same d − 3 variables to each vertex to show µ(d, d + 3) 3.
To see that Ḡ(R) is locally connected, we take any subset of variables S and check that the vertices containing S form a connected subgraph.Below, we color the variables containing E red.Proof.We can prove that there does not exist Ḡ(R), such that R is an (S 2 ) Stanley-Reisner ring, n = 9, d = 3, and diam( Ḡ(R)) > 7. Due to length, this proof has been omitted from the paper but can be found at: http://www.math.ku.edu/ ~b101h187/.The graph in Figure 6 with the vertex EHI appended is a diameter-7 graph with properties (i) and (ii) when n = 9, d = 3.
Proposition 41 gives a bound only one better than the general upper bound given in Theorem 19.
Proof.Proof.We can construct an example of a diameter-(n − d + 2) graph with properties (i) and (ii) with d = 3, n = 10+j by taking the graph in Figure 7 and appending the vertices: Buchsbaum complexes have long been studied in combinatorial algebra [13,25,11,24].It is of interest that all of the complexes we have examined thus far are Buchsbaum.The following is likely known to experts, but we include a proof here.
Proposition 44.Let R = S/I be an equidimensional Stanley-Reisner ring of dimension 3. Then R is connected and Buchsbaum if and only if R satisfies (S 2 ).
Proof.Let ∆ be a complex with Stanley-Reisner ring R = S/I.Let m be the unique homogeneous maximal ideal of S. Using local duality (see Lemma 27) R) is finitely generated for all i < d.Thus for R an equidimensional Stanley-Reisner ring of dimension 3, Buchsbaum and connected imply (S 2 ).
Suppose now that R satisfies (S 2 ).We will use the combinatorial definition of Buchsbaum, which says a complex is Buchsbaum if it is pure and has the property that the link of any non-empty face has zero reduced homology except possibly in top dimension.
We first note the Stanley-Reisner ring of ∆ satisfying (S 2 ) implies ∆ is pure and connected.Next we note that every link of a non-empty face of ∆ has Stanley-Reisner ring R P , where P is a prime ideal and dim R P < dim R. Thus R being a 3-dimensional, (S 2 ) ring implies R P is Cohen-Macaulay for all P such that dim R P < dim R. Thus R P has zero reduced homology except possibly in top dimension.Thus R is Buchsbaum and connected.
Note that this theorem does not apply in the higher dimension cases.In fact, most of our examples in higher dimension, including Figure 8  Proof.Let G be a graph with properties (i) and (ii) and diameter at least 8. First let us consider two vertices with maximum distance in G.If the intersection of these vertices is non-trivial, their distance is bounded above by µ(3, 8) = 6.Thus, the vertices of maximal distance in G must have trivial intersection.Call them ABCD and F GHI. Suppose there exists a v ∈ V (G) such that v is adjacent to ABCD, and v contains F, G, H or I.
The shortest path from v to F GHI will be bounded above by µ(3, 8) = 6, and thus G will have diameter at most 7. Thus no such vertex is contained in G. Simillarly no vertex containing A, B, C or D adjacent to F GHI is contained in G.
Connectivity of G requires G have at least one vertex adjacent to ABCD and at least one vertex adjacent to F GHI.These vertices must both contain the only variable which is not in ABCD or F GHI, call this variable E. Since G is locally connected, G must contain a connected subgraph composed only of the vertices containing E. Every vertex containing E will also contain two variables from either ABCD or F GHI. Take such a vertex, ABEF .Then G must have a connected subgraph made up of only the vertices containing AB.However, we already have that any vertex adjacent to ABCD must contain E. Thus ABEF must have shortest path length 2 to ABCD.Any vertex the electronic journal of combinatorics 25(1) (2018), #P1.60 containing E is distance at most 2 from ABCD or F GHI. Thus G is at most diameter 5. Thus we have a contradiction.Thus no diameter-8 graph with properties (i) and (ii) exists.
To construct a diameter-7 graph with properties (i) and (ii), take the graph in Figure 8 and append the vertex EF GI.Proof.We can construct an example of a diameter-(n − d + 2) graph with propeties (i) and (ii) with d = 4, n = 8 + j by taking the graph in Figure 8 and appending the vertices EF Gx 1 , F Gx 1 x 2 , . . ., x j−3 x j−2 x j−1 x j .

2 Background and Notation 2 . 1 Definition 3 .
Introduction of Terms Definition 1.A d-dimensional polyhedron is a non-empty intersection of finitely many closed half spaces of R d .Definition 2. A facet of a simplicial complex ∆ is a simplex of ∆ which is not properly contained in another simplex of ∆.The 1-skeleton of a polyhedron is the set of vertices and edges of the polyhedron.

2 .
G is Ḡ(R) with R an equidimensional Stanley-Reisner ring, and G has local connectedness.3. G is Ḡ(R) with R an (S 2 ) Stanley-Reisner ring.4. G is Ḡ(R) with R = S/I an equidimensional Stanley-Reisner ring such that the Alexander dual of I, I ∨ , has first syzygy matrix with all linear entries.the electronic journal of combinatorics 25(1) (2018), #P1.60 Define a block of layers to be a set of layers {L i |a i b} for some integers a, b.Let c be the largest integer such that L c contains A, B, or C. Let B 1 = {L i |0 i c}.Let n 0 be the number of variables in B 1 .Without loss of generality, A is contained in L c .By local connectedness, there exists a path consisting of vertices which all contain A from L 0 to L c .This path can have maximum length n 0 − 3 (see the d = 2, n = n 0 − 1 case in Appendix A).
Thus the path from ABC to v * must have length at most n − 7. The maximum length from v * to DEF is n − 3. Thus, L D 2n − 10.Consider the case whereL A = n − 4. Take v ∈ L n−4 such that A ∈ vand construct a path from ABC to v. If the path contains AD, AE, and AF , then the path from ABC to v * must have length at most n − 7. If this path does not contain one of those, say AF , then i * n − 6.If i * = n − 6, then L A L B L C implies any strictly increasing path from v * to DEF of vertices all containing D cannot have both a vertex containing B and a vertex containing C. Thus L D i * + n − 4 = 2n − 10.Consider i * < n − 6.Any path from v * to DEF will be at most length n − 3. Thus L D 2n − 10.Now suppose L A n − 5. Then i * n − 5. Further, B, C / ∈ L j for all j > n − 5. Thus any path from L A to DEF in which each vertex contains D, will have maximal length n − 3 − 2. Thus L D 2n − 10.Definition 20. µ(4, 8) 6. Proof.Let us consider v 1 , v 2 ∈ Ḡ(R).If v 1 ∩ v 2 = ∅ then we can reduce to the n = 7, d = 3 case, and thus dist(v 1 , v 2 ) 5. Thus assume v 1 ∩ v 2 = ∅.Connectivity implies there exists v 3 adjacent to v 1 .The vertex v 3 must have a non-trivial intersection with v 2 , thus dist(v 2 , v 3 ) 5. Therefore, dist(v 1 , v 2 ) 6. Definition 21.For d 4, µ(d, d + 4) = 6.
Theorem 31.µ(3, 8k + 2) 10k − 1.Proof.Construct a graph composed of k − 1 copies of G 2 by gluing ABC to IJK.Then glue a copy of G 1 to the end.Each copy of G 2 adds 10 to diameter, since all adjacent vertices lie in the same copy of G 2 .Gluing G 1 adds 9 to the diameter.The new graph retains local connectedness byTheorem 28.This yields a lower bound of5  4 (n − d) when n = 8k + 2, d = 3. Definition 32.µ(3, 8k + 3 + j) 10k + j when j 0.

Figure 7
is an example of a diameter-9 graph with properties (i) and (ii) with d = 3, n = 10.Proposition 42 gives a bound only one better than the bound of Theorem 19.Proposition 43.µ(3, n) n − d + 2 for all n 10.

Table 1 :
Introduction of terms is covered in Section 2.1.In Section 2.2, we demonstrate that a graph having properties (i) and (ii) is equivalent to that graph being the dual graph of a Stanley-Reisner ring satisfying Serre's condition (S 2 ).In Section 3, we prove the upper bounds introduced earlier in this section.Details of the process of gluing to preserve (S 2 ) are discussed in Section 4, and constructions of glued complexes are discussed in Section 5 with examples displayed in Figures1 and 2. In Appendix A, we show the constructions needed which justify Table1and investigate the relationship between (S 2 ) and Buchsbaum.
2 ) Stanley-Reisner ring of dimension d and codimension n − d.Theorem 35.Table 1 (see Introduction) presents µ(d, n) for small values of d and n.
, we have that a 3-dimensional equidimensional Stanley-Reisner ring R satisfies (S 2 ) if and only if dim Ext n−i S (R, ω S ) i − 2 for all i < 3. Thus, R satisfies (S 2 ) if and only if H 0 m