DERIVING SOME PROPERTIES OF STANLEY-REISNER RINGS FROM THEIR SQUAREFREE ZERO-DIVISOR GRAPHS

Let ∆ be a simplicial complex, I∆ its Stanley-Reisner ideal and R = K[∆] its Stanley-Reisner ring over a field K. In 2018, the author introduced the squarefree zero-divisor graph of R, denoted by Γsf(R), and proved that if ∆ and ∆′ are two simplicial complexes, then the graphs Γsf(K[∆]) and Γsf(K[∆ ′]) are isomorphic if and only if the rings K[∆] and K[∆′] are isomorphic. Here we derive some algebraic properties of R using combinatorial properties of Γsf(R). In particular, we state combinatorial conditions on Γsf(R) which are necessary or sufficient for R to be Cohen-Macaulay. Moreover, we investigate when Γsf(R) is in some well-known classes of graphs and show that in these cases, I∆ has a linear resolution or is componentwise linear. Also we study the diameter and girth of Γsf(R) and their algebraic interpretations. Mathematics Subject Classification (2020): 13F55, 13C70, 05C25, 05E40


Introduction
In this paper all rings are commutative with identity and K is a field. Let S = K[x 1 , . . . , x n ] be the polynomial ring in n indeterminates over K. By a squarefree monomial ideal of S we mean an ideal generated by a set of squarefree monomials of S. In the last few decades, the study of squarefree monomial ideals has got a large attention (for example, see [3, 7-9, 11-14, 16]). This is because of the fact that if we know algebraic properties of squarefree monomial ideals well, then we can understand many algebraic properties of much larger classes of ideals such as graded ideals of S (see [7]).
There are strong relations between squarefree monomial ideals and several combinatorial objects (see, for instance, [8,11,12] and Part III of [7]). Here we use two related combinatorial objects. The first one is the concept of simplicial complexes. On the other hand, recently several authors have defined some graphs based on the structure of rings and used them to study the algebraic properties of these rings. One of the first defined and most studied of such graphs is the zero-divisor graph, see for example [1,6,10,15] and the references therein. It is quite usual that these graphs, which are defined for general commutative rings, are isomorphic for non-isomorphic rings, see for example Theorem 2.2 and Corollary 2.4 of [12].
In [12], the author used the ideas in the definition of zero-divisor graphs, in the special case of Stanley-Reisner rings and introduced squarefree zero-divisor graphs of Stanley-Reisner rings as a "stronger version" of the zero-divisor graphs of such rings. Suppose that I is a squarefree monomial ideal of S and R = S/I. Let V be image in R of the set of all squarefree monomials of S which are not in I. By the squarefree zero-divisor graph of R, we mean the graph on vertex set V , in which, two vertices u and v are adjacent if and only if uv = 0 in R. We denote this graph by Γ sf (R). If I = I ∆ for a simplicial complex ∆, we also use Γ sf (∆) instead of Γ sf (R).
Note that Γ sf (∆) can be described In the sequel, all graphs are finite, simple and undirected. If u and v are vertices of a graph, by v ∼ u we mean that u and v are adjacent. We denote the set of vertices of a graph G by V(G). Any undefined graph theoretic notation is as in [17]. Also recall that the maximal faces of a simplicial complex ∆ are called facets and by ∆ = ⟨F 1 , . . . , F t ⟩ we mean that F 1 , . . . , F t are all facets of ∆. Moreover, dim ∆ = max{|F |−1|F ∈ ∆}. For more on (squarefree) monomial ideals, simplicial complexes, and the related algebraic concepts, such as linear resolutions or Cohen-Macaulayness, see [2,7].

Combinatorial properties necessary for Cohen-Macaulayness
One of the important topics of research in commutative algebra is finding characterizations of Cohen-Macaulay commutative rings. In particular, many have tried to give combinatorial characterizations for Cohen-Macaulayness of certain classes of squarefree monomial ideals (see for example [7,8]). It is well-known that if I ∆ is Cohen-Macaulay and dim ∆ > 0, then ∆ is pure (that is, all of its facets have the same size) and connected (which means there is a sequence of faces F 1 , . . . , F t between any two nonempty faces F 1 and F t , such that F i ∩ F i+1 ̸ = ∅). Here we study when ∆ is pure or connected in terms of Γ sf (∆). In the following, by G we mean the complement of a graph G. Also we always assume that ∆ is a simplicial complex on [n], unless stated otherwise explicitly. Proof. Suppose that ∆ is connected. If x F and x G are two vertices of Γ sf (∆), then there is a sequence F = F 1 , . . . , F t = G of faces of ∆ such that F i ∩ F i+1 ̸ = ∅, for each i. Now in Γ sf (∆), we have the following path between x F and x G : Conversely, assume that Γ sf (∆) is connected. If ∆ is not connected, then there is a partition [n] = V 1 ∪ V 2 such that no face of ∆ has vertices from both V 1 and V 2 . if there is a nonnegative integer r such that each vertex of Γ sf (∆) which is in a largest clique has degree r.
Proof. First suppose that F is a facet of ∆ and A is an arbitrary face. Then if and only if x A ̸ ∼ x F . Therefore the set all vertices not adjacent to x F correspond exactly to all nonempty subsets of F , the number of which is 2 |F | − 1. Therefore deg( form. Also note that, by the proof of Theorem 2.2, x G F and x F have the same neighborhoods.
As instant corollaries of the above results we get the following.
Corollary 2.4. Suppose that I is a squarefree monomial ideal and R = S/I. If R is Cohen-Macaulay, then either Γ sf (R) is a complete graph or Γ sf (R) is connected and the degrees of all vertices of Γ sf (R) contained in a largest clique are equal.
Therefore we can assume that I ⊆ ⟨x 1 , . . . , x n ⟩ 2 , hence I = I ∆ for a simplicial complex ∆. If dim ∆ = 0, then Γ sf (∆) is a complete graph. Else, since R is Cohen- Next we study the diameter of squarefree zero-divisor graphs. Recall that the diameter of a graph G, denoted diam G, is the maximum distance of two vertices Let's denote the latter graph byΓ sf (R). If ∆ has exactly one facet, then R is a polynomial ring over K and Γ sf (R) is just a set of isolated vertices. In other words,Γ sf (R) is empty in this case. The following result shows thatΓ sf (R) is the "main part" of Γ sf (R), in the sense that if we knowΓ sf (R), then we can reconstruct Γ sf (R), unlessΓ sf (R) = ∅. Proof. Let R = K[∆] and ∆ = ⟨F 1 , . . . , F t ⟩. We know that Γ sf (R) is a disjoint union ofΓ sf (R) and a set J of 2 |F0| − 1 isolated vertices, where F 0 = ∩ t i=1 F i . Suppose that I is a maximal independent set ofΓ sf (R) meeting a largest clique of Γ sf (R). Then this clique is also a largest clique in Γ sf (R) and hence I ′ = I ∪ J is a maximal independent set of Γ sf (R) meeting a largest clique of Γ sf (R). According to [12,Corollary 3.6], in [4,5]. The vertices of this graph are nonzero zero-divisors of E and two vertices x and y are connected when xy = 0. Set E = V(Γ sf (R)) ∪ {0} and define 0 · x = 0 for all x ∈ E and is a commutative semigroup with zero and Γ(E) =Γ sf (R). We get the following as an immediate corollary of this fact.  (ii) Assume that diamΓ sf (R) ≤ 2 and x i and x j are distinct indeterminates appearing in some minimal generators of I such that x i x j / ∈ I. Since x i and x j appear in some minimal generators of I, x i , x j ∈ Z(R) and hence x i , x j ∈Γ sf (R).
Since x i x j ̸ = 0, these vertices are not adjacent and as diamΓ sf By replacing F with a facet of ∆ containing F , we can assume that F is a facet of ∆.

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According to [7,Lemma 1.5.4], PF is a minimal prime of I and x i , x j ∈ PF .
Suppose that (I : x i ) ∩ (I : x j ) ⊆ I + ⟨x i , x j ⟩ ⊆ PF . Then either (I : x i ) ⊆ PF or (I : x j ) ⊆ PF . Without loss of generality, assume that the former holds. Let P ′ = PF \{i} which is strictly contained in PF . So by the minimality of PF , we have To complete the proof we get a contradiction by showing that I ⊆ P ′ . Let u ∈ G(I). If x i ∤ u, then as u ∈ PF , x r |u for some i ̸ = r ∈F and hence u ∈ P ′ .
If x i |u, then u/x i ∈ (I : x i ) ⊆ PF and x i ∤ (u/x i ). Thus as above, u/x i ∈ P ′ and hence u ∈ P ′ . This shows that I ⊆ P ′ as required.
Conversely, assume that for each pair of distinct indeterminates x i and x j appearing in some minimal generator of I such that x i x j / ∈ I, we have (I : x i ) ∩ (I : be two nonadjacent vertices ofΓ sf (R). We must find a vertex adjacent to both x F1 and x F2 . If F 1 ⊆ F 2 , then every vertex adjacent to x F1 is also adjacent to x F2 . So in this case, sinceΓ sf (R) has not isolated vertices, we are done. Thus we assume that F 1 and F 2 are incomparable. If i ∈ F 1 \ F 2 and j ∈ F 2 \ F 1 , then x i and x j are nonadjacent and every vertex adjacent to both x i and x j is also adjacent to both x F1 and x F2 . So we can assume that F 1 = {i} and By assumption, (I : x i ) ∩ (I : x j ) ̸ ⊆ I + ⟨x i , x j ⟩ and as I + ⟨x i , x j ⟩ is squarefree monomial and hence a radical ideal, there is a minimal prime P F of I + ⟨x i , x j ⟩ such that (I : Proof. If ∆ has at least three different facets x F1 is a cycle with length 3. If ∆ has one facet, then Γ sf (∆) is a set of isolated vertices. If ∆ has two facets F 1 , F 2 , then Γ sf (∆) is disjoint union of a set of 2 |F1∩F2| − 1 isolated vertices and a complete bipartite graph with partition sizes 2 |F1| −2 |F1∩F2| and 2 |F2| − 2 |F1∩F2| . From this, the result easily follows. □

When Γ sf (R) is in a well-known class of graphs
In the final section of this article, we investigate when Γ sf (R) is in some of the well-known classes of graph. Recall that a graph is complete r-partite, when its set of vertices can be partitioned to r subsets such that two vertices are adjacent if and only if they are from different subsets. Moreover, a graph is called regular, when all of its vertices have the same degree. Also a graph G is said to be chordal, when for each S ⊆ V(G), the induced subgraph of G on S is not a cycle, unless |S| = 3.
(iv) Γ sf (R) is regular, if and only if there are positive integers n, r such that Proof. Throughout the proof we assume that R = K[∆] where ∆ = ⟨F 1 , . . . , F t ⟩.
(i) Indeed, this was proved in the proof of Theorem 3.3(i).
(ii) Suppose Γ sf (R) is bipartite. Then t = |Ass(R)| equals the size of the largest clique of Γ sf (R) by [12,Corollary 3.5]. Since Γ sf (R) is bipartite, it follows that t ≤ 2. If t = 1, then R has the claimed form with a = |F 1 | and n = m = 0.
. . , n + m} and is the ring claimed in the statement. Conversely, suppose that R has the specified form. Then it is easy to see that in Γ sf (R) the set of vertices , 1), . . . , (i, n i )} is independent for each i and if i ̸ = j then every vertex of P i is adjacent to every vertex of P j . Therefore, Γ sf (R) is complete r-partite.
(iv) If R is the specified ring, then by part (iii), Γ sf (R) is a complete r-partite graph in which all parts have the same size. Thus Γ sf (R) is regular. Conversely, (v) Suppose that R is any of the two rings mentioned in this part. It is routine to see that vertices of Γ sf (R) can be partitioned into two sets V 1 and V 2 , such that V 1 is a clique and V 2 is an independent set of Γ sf (R). Therefore any induces cycle of Γ sf (R) with length at least 4, has at most 2 vertices in V 1 . Therefore, such a cycle must have at leas two adjacent vertices in V 2 , a contradiction. Hence Γ sf (R) is chordal.
and V 2 = [n] \ V 1 . We can assume that V 1 = [r] for some r ≤ n. Suppose that Thus there is a squarefree monomial u ∈ G(I) \ I 0 . If deg(u) = 2, then u = x i x j for some r < i ̸ = j ≤ n. Since i, j > r, Suppose that deg(u) ≥ 4. Then u = x i x j x k x l v for some squarefree monomial v. Now we have the induced cycle , then x F ∈ G(I) for each F ⊆ {r + 1, . . . , n} such that |F ∩ {i, j, k}| = 2. We just need to show that x F ∈ I, for such sets F . Suppose x F / ∈ I for some such F , say F = {i, j, l} with l / ∈ {i, j, k}. Then we get the following induced cycle of length 4 which is a contradiction: x k . Now assume that G = {a, b, c} is an arbitrary 3-subset of {r + 1, . . . , n}. If |G ∩ {i, j, k}| = 1, say a = i, then by the above argument x i x j x c ∈ I and hence again applying the above argument with {i, j, c} instead of {i, j, k} we see that x G ∈ G(I). If G ∩ {i, j, k} = ∅, then by a similar argument we see that x G ∈ G(I). Thus Recall that a graded ideal I of S has a linear resolution if it can be generated in degree d and its Castelnuovo-Mumford regularity (see definition (2)  for all monomials u ∈ I (or equivalently, u ∈ G(I)) and for all j < m(u) such that x j does not divide u, one has x j (u/x m(u) ) ∈ I, where m(u) denotes the largest index of an indeterminate which divides u. is complete.
(v) Suppose that Γ sf (R) is chordal. Then R is sequentially Cohen-Macaulay.
Also R is Cohen-Macaulay if and only if either Γ sf (R) or Γ sf (R) is complete or the set of vertices of Γ sf (R) can be partitioned into an independent set V 1 with size n and a clique V 2 with size n 2 such that for each vertex , then by Theorem 4.1(i), ∆ is zero-dimensional and hence Cohen-Macaulay.
(iii) Let G be the graph in which vertices denote the indeterminates of S and two vertices are adjacent when their product is in I. Then it follows Theorem 4.1(iii) that G is a complete r-partite graph. Suppose that part i has size n i with 1 ≤ n 1 ≤ n 2 · · · ≤ n r . By [ If r = n, then Γ sf (R) is complete. If r = 0, then either ∆ has exactly one facet and Γ sf (R) is a set of isolated vertices or ∆ = ⟨{i, j}|1 ≤ i < j ≤ n⟩. In the latter case, V(Γ sf (R)) = V 1 ∪ V 2 with V 1 = {x 1 , . . . , x n } and V 2 = {x F |F ⊆ [n], |F | = 2}. Also V 1 is an independent set and V 2 is a clique and each vertex x i of V 1 is adjacent to all x F ∈ V 2 with i / ∈ F . So deg(x i ) = |V 2 | − (n − 1).
Conversely, if such a partition of V(Γ sf (R)) exists, then Γ sf (R) is chordal and ∆ is one of the aforementioned complexes. Because there does not exist any vertex adjacent to all other vertices, we must have r = 0 and ∆ is pure. From this, the claim follows. □