Rees algebra and special fiber ring of binomial edge ideals of closed graphs
aa r X i v : . [ m a t h . A C ] F e b REES ALGEBRA AND SPECIAL FIBER RING OF BINOMIAL EDGEIDEALS OF CLOSED GRAPHS
ARVIND KUMAR
Abstract.
In this article, we compute the regularity of Rees algebra of binomial edge idealsof closed graphs. We obtain a lower bound for the regularity of Rees algebra of binomialedge ideals. We also study some algebraic properties of the Rees algebra and special fiberring of binomial edge ideals of closed graphs via algebraic properties of their initial algebraand Sagbi basis theory. We obtain an upper bound for the regularity of the special fiberring of binomial edge ideals of closed graphs. Introduction
Let S be a standard graded polynomial ring over a field K . Let I be a homogenous idealin S . The K -subalgebra ⊕ n ≥ I n t n ⊂ S [ t ] is known as the Rees algebra of I , and is denoted by R ( I ). The Rees algebra of a homogeneous ideal encodes a lot of asymptotic properties ofthat ideal. In this paper, we study the Rees algebra of binomial edge ideals of closed graphs.An ideal generated by a set of 2-minors of a 2 × n generic matrix is known as binomial edgeideal . These ideals were introduced by Herzog et al. in [13] and independently by Ohtaniin [25] a decade ago. Let G be a simple graph with vertex set V ( G ) = [ n ] := { , . . . , n } and edge set E ( G ). The binomial edge ideal of G is defined as J G = ( x i y j − x j y i : i Binomial edge ideal, Rees Algebra, special fiber ring, Castelnuovo-Mumfordregularity, Closed graphs.AMS Subject Classification (2010): 13D02, 13C13, 13A13, 05E40. algebraic properties of the Rees algebra of binomial edge ideals of closed graphs via algebraicproperties of its initial algebra and Sagbi basis theory.Another K -algebra associated with a homogeneous ideal is the special fiber ring. The special fiber ring of a homogeneous ideal I is the ring F ( I ) = R ( I ) / m R ( I ) ∼ = ⊕ k ≥ I k / m I k ,where m is the homogeneous maximal ideal of S . Nothing much is known about the specialfiber ring of binomial edge ideals. The author in [20] proved that the special fiber ring ofbinomial edge ideals of forests is a polynomial ring. In this paper, we study the specialfiber ring of binomial edge ideals of closed graphs. We prove that the special fiber ring ofbinomial edge ideals of closed graphs is Koszul and Cohen-Macaulay normal. We obtain theanalytic spread of binomial edge ideals of closed graphs in terms of the number of verticesand number of indecomposable components of G . We also obtain an upper bound for theregularity of the special fiber ring of binomial edge ideals of closed graphs.The article is organized as follows. The second section contains all the necessary definitionsand notation required in the rest of the article. In Section 3, we study the Rees algebra ofbinomial edge ideals of closed graphs. We study the special fiber ring of binomial edge idealsof closed graphs in Section 4. 2. Preliminaries In this section, we collect all the notions that we will use in this paper. We first recall allthe necessary definitions from graph theory.Let G be a simple graph with the vertex set [ n ] and edge set E ( G ). A graph on [ n ] issaid to be a complete graph , if { i, j } ∈ E ( G ) for all 1 ≤ i < j ≤ n . The complete graphon [ n ] is denoted by K n . For A ⊆ V ( G ), G [ A ] denotes the induced subgraph of G on thevertex set A , that is, for i, j ∈ A , { i, j } ∈ E ( G [ A ]) if and only if { i, j } ∈ E ( G ). A subset U of V ( G ) is said to be a clique if G [ U ] is a complete graph. A vertex is said to be a simplicial vertex if it belongs to exactly one maximal clique. The clique number of a graph G , denoted by ω ( G ), is the maximum size of the maximal cliques of G . For a vertex v , N G ( v ) = { u ∈ V ( G ) : { u, v } ∈ E ( G ) } denotes the neighborhood of v in G . The degree of avertex v , denoted by deg G ( v ), is | N G ( v ) | . A cycle is a connected graph G with deg G ( v ) = 2for all v ∈ V ( G ). A graph is a tree if it does not have a cycle. A graph is said to be a forest if each connected component is a tree. A graph G is said to be bipartite if there is abipartition of V ( G ) = V ⊔ V such that for each i = 1 , 2, no two of the vertices of V i areadjacent. A graph is called a non-bipartite graph if it is not a bipartite graph. A subset M of E ( G ) is said to be a matching of G if e ∩ e ′ = ∅ for every pair e, e ′ ∈ M with e = e ′ . The matching number of a graph G , denoted by mat( G ), is the maximum size of the maximalmatchings of G . A matching M is said to be a perfect matching if V ( G ) = ∪ e ∈ M e . For agraph H , a graph G is said to be H -free graph if H is not induced subgraph of G .We recall the notation of decomposability from [26]. A graph G is called decomposable , ifthere exist subgraphs G and G such that G is obtained by identifying a simplicial vertex v of G with a simplicial vertex v of G , i.e., G = G ∪ G with V ( G ) ∩ V ( G ) = { v } such that v is a simplicial vertex of both G and G . A graph G is called indecomposable , ifit is not decomposable. Up to ordering, G has a unique decomposition into indecomposablesubgraphs, i.e., there exist G , . . . , G r indecomposable induced subgraphs of G with G = G ∪ · · · ∪ G r such that for each i = j , either V ( G i ) ∩ V ( G j ) = ∅ or V ( G i ) ∩ V ( G j ) = { v } and v is a simplicial vertex of both G i and G j . EES ALGEBRA AND SPECIAL FIBER RING OF BINOMIAL EDGE IDEALS OF CLOSED GRAPHS 3 Now, we recall all the necessary notation from commutative algebra. Let R = K [ x , . . . , x m ]be a standard graded polynomial ring over an arbitrary field K and M be a finitely generatedgraded R -module. Let0 −→ M j ∈ Z R ( − j ) β Rp,j ( M ) φ p −→ · · · φ −→ M j ∈ Z R ( − j ) β R ,j ( M ) φ −→ M −→ , be the minimal graded free resolution of M , where R ( − j ) is the free R -module of rank 1generated in degree j . The number β Ri,j ( M ) is called the ( i, j )-th graded Betti number of M . The Castelnuovo-Mumford regularity(simply regularity) of M , denoted by reg( M ), isdefined as reg( M ) := max { j − i : β Ri,j ( M ) = 0 } . Rees Algebra of binomial edge ideals of closed graphs In this section, we study the regularity of Rees algebra of binomial edge ideals of closedgraphs. We also study some other algebraic properties of R ( J G ) via algebraic properties of R (in σ ( J G )) and Sagbi basis theory.Let K be a field and S = K [ x , . . . , x n ] be a standard graded polynomial ring over K . Let A be a finitely generated K − subalgebra of S generated by homogeneous elements. Let σ bea term order for the monomials in S . The initial algebra of A with respect to the term order σ is the K − subalgebra of S generated by { in σ ( f ) : f ∈ A } . Let A i denote the homogeneouscomponent of degree i . The K − vector space spanned by { in σ ( f ) : f ∈ A i } is denoted byin σ ( A i ). It follows from [4, Proposition 2.4] thatin σ ( A ) = ⊕ i ≥ in σ ( A i ) . Moreover, the Hilbert functions of A and in σ ( A ) concide.Let I be a homogeneous ideal in S . The Rees algebra of I is defined as R ( I ) = ⊕ i ≥ I i t i . Throughout this article, we assume that I is an equi-generated homogeneous ideal. Let { f , . . . , f m } be a minimal homogeneous generating set of I , and R = S [ t , . . . , t m ] be astandard graded polynomial ring over K . Let ψ : R → S [ t ] be the S -algebra homomorphismgiven by ψ ( t i ) = f i t . Then, R/ ker( ψ ) ≃ Im( ψ ) = R ( I ), where ker( ψ ) is a homogeneousideal of R and it is called the defining ideal of R ( I ). The relation type of an ideal I is thelargest t -degree of a minimal generator of the defining ideal of R ( I ). We say that I is of linear type if Rees algebra of I is isomorphic to the symmetric algebra of I . If the definingideal of Rees algebra is generated by defining equations of the symmetric algebra and thedefining equations of the special fiber ring, then we say that I is of fiber type .Let σ be a term order for the monomials in S . Let τ be term order for monomials in S [ t ] defined as follows: given two monomials u, v ∈ S and two integers i, j ≥ , we have ut i < τ vt j if and only if i < j or i = j and u < σ v. Then, the initial algebra of R ( I ) withrespect to the term order τ is in τ ( R ( I )) = ⊕ i ≥ in σ ( I i ) t i . Notation 3.1. Let I be an equi-generated homogeneous ideal in S . Then, by reg( R ( I )) , wemean the regularity of R/ ker( ψ ) as R -graded module. ARVIND KUMAR Theorem 3.2. Let I = ( f , . . . , f m ) ⊂ S be an equi-generated homogeneous ideal whichsatisfies the followings:(1) in σ ( I ) = (in σ ( f ) , . . . , in σ ( f m )) ;(2) in σ ( I s ) = (in σ ( I )) s , for all s ≥ ;(3) R (in σ ( I )) is Cohen-Macaulay.Then, reg( R ( I )) = reg( R (in σ ( I ))) .Proof. Since in σ ( I s ) = (in σ ( I )) s , for all s ≥ 1, it follows from [4, Theorem 2.7] thatin τ ( R ( I )) = R (in σ ( I )). Thus, it is enough to prove that reg( R ( I )) = reg(in τ ( R ( I ))). Sincein τ ( R ( I )) is Cohen-Macaulay, by [4, Corollary 2.3], R ( I ) is Cohen-Macaulay, and Krulldimension of R ( I ) and in τ ( R ( I )) are same. It follows from [4, Proposition 2.4] thatHS R ( I ) ( λ ) = HS in τ ( R ( I )) ( λ ) , and therefore, h R ( I ) ( λ ) = h in τ ( R ( I )) ( λ ) , where h R ( I ) ( λ ) is the h -polynomial of of R ( I ). For a standard graded Cohen-Macaulayalgebra S/J , it is well known that reg( S/I ) = deg h S/I ( λ ), (see [7, Section 4.1]). Therefore,reg( R ( I )) = deg h R ( I ) ( λ ) and reg(in τ ( R ( I ))) = deg h in τ ( R ( I )) ( λ ). Hence,reg( R ( I )) = reg(in τ ( R ( I ))) = reg( R (in σ ( I )))which proves the assertion. (cid:3) Let G be a graph on [ n ] and J G be its binomial edge ideal in the standard graded polyno-mial ring S = K [ x , . . . , x n , y , . . . , y n ]. Let σ be the lexicographic term order on S inducedby x > · · · > x n > y > · · · > y n . A graph G is said to be closed with respect to givenlabelling of vertices if the generators of J G form a quadratic Gr¨obner basis of J G with respectto the term order σ . We say that a graph is a closed graph if it is closed with respect tosome labelling of vertices. Set R = S [ T { i,j } : { i, j } ∈ E ( G ) with i < j ] to be standardgraded poynomial ring over K .We now compute the regularity of Rees algebra of binomial edge ideals of closed graphs. Theorem 3.3. Let G be a closed graph on [ n ] with respect to given labelling of vertices.Assume that G has no isolated vertices. Then, reg( R ( J G )) = reg( R (in σ ( J G ))) = n − c, where c is the number of connected components of G .Proof. Since G is a closed graph with respect to given labelling of vertices, by [13, Theorem1.1], in σ ( J G ) = ( x i y j : i < j, { i, j } ∈ E ( G )). It follows from the proof of [8, Lemma 3.1] thatin σ ( J sG ) = (in σ ( J G )) s , for all s ≥ 1. By [10, Proposition 2.9], R (in σ ( J G )) is Cohen-Macaulay.Thus, by Theorem 3.2, reg( R ( J G )) = reg( R (in σ ( J G ))).First, we assume that G is connected. Let H be a graph on the vertex set { x . . . , x n − } ⊔{ y , . . . , y n } and edge set {{ x i , y j } : i < j, { i, j } ∈ E ( G ) } . It follows from [11, Lemma 3.3]that H is a bipartite graph and the monomial edge ideal of H is I ( H ) = in σ ( J G ). By [9,Section 2], { i, i + 1 } ∈ E ( G ) for 1 ≤ i ≤ n − 1, therefore {{ x i , y i +1 } : 1 ≤ i ≤ n − } ⊂ E ( H )is a perfect matching of H . Therefore, by [3, Theorem 4.2], reg( R ( I ( H ))) = mat( H ) = n − H ) is the matching number of H . Thus, reg( R ( J G )) = reg( R (in σ ( J G ))) = n − . Now, assume that G is not connected. Let G , . . . , G c be connected components of G .For each 1 ≤ k ≤ c , let H k be the bipartite graph such that I ( H k ) = in σ ( J G k ). Then, EES ALGEBRA AND SPECIAL FIBER RING OF BINOMIAL EDGE IDEALS OF CLOSED GRAPHS 5 H = H ⊔ · · · ⊔ H k is a bipartite graph with a perfect matching of size n − c . By virtueof [3, Theorem 4.2], we have reg( R ( I ( H ))) = mat( H ) = n − c . Hence, reg( R ( J G )) =reg( R (in σ ( J G ))) = n − c. (cid:3) Corollary 3.4. Let G be a closed graph on [ n ] . Assume that G has no isolated vertices.Then, reg( R ( J G )) = n − c , where c is the number of connected components of G . Now, we obtain a lower bound for the regularity of Rees algebra of binomial edge ideals. Theorem 3.5. Let G be a graph on [ n ] . If H is an induced subgraph of G , then reg( R ( J H )) ≤ reg( R ( J G )) . In particular, ℓ ( G ) ≤ reg( R ( J G )) , where ℓ ( G ) is the length of a longest induced path in G .Proof. Let H be an induced subgraph of G . It follows from the proof of [16, Proposition 3.3]that J sH = J sG ∩ S H , where S H = K [ x j , y j : j ∈ V ( H )]. Therefore, for every i ≥ J iH t i = J iG t i ∩ S H [ t ]. Then, R ( J G ) ∩ S H [ t ] = ( ⊕ i ≥ J iG t i ) ∩ S H [ t ] = ⊕ i ≥ (cid:0) J iG t i ∩ S H [ t ] (cid:1) = ⊕ i ≥ J iH t i = R ( J H ).Thus, R ( J H ) is a K -subalgebra of R ( J G ). Set R H = S H [ T { i,j } : { i, j } ∈ E ( H )]. Let I and I be ideals of R and R H , respectively such that R/I ≃ R ( J G ), and R H /I ≃ R ( J H ). Now,define π : R/I → R H /I as π ( x j ) = π ( y j ) = 0 if j V ( H ), π ( x j ) = x j , π ( y j ) = y j if j ∈ V ( H ), π ( T { i,j } ) = 0 if { i, j } 6∈ E ( H ), and π ( T { i,j } ) = T { i,j } if { i, j } ∈ E ( H ). Consider, R H /I i ֒ → R/I π −→ R H /I . Then, π ◦ i is identity on R H /I , and hence, R H /I is an algebraretract of R/I . It follows from [22, Corollary 2.5] that reg( R H /I ) ≤ reg( R/I ). Hence,reg( R ( J H )) ≤ reg( R ( J G )).Let H be a longest induced path of G . Then, H is an induced path of G . Since H is a closedgraph, by Corollary 3.4, reg( R ( J H )) = | V ( H ) | − ℓ ( G ). Hence, reg( R ( J G )) ≥ ℓ ( G ). (cid:3) We now move on to study a Sagbi basis for R ( J G ), and using that we study some ofalgebraic properties of R ( J G ) via algebraic properties of in τ ( R ( J G )). Theorem 3.6. Let G be a closed graph on [ n ] with respect to given labelling of vertices.Then, in τ ( R ( J G )) = R (in σ ( J G )) and reltype ( J G ) ≤ . Moreover, the set { x i , y i : 1 ≤ i ≤ n } ∪ { f e t : e ∈ E ( G ) } is a Sagbi basis of R ( J G ) with respect to term order τ .Proof. Let H be the graph constructed in the proof of Theorem 3.3, i.e. I ( H ) = in σ ( J G ).By the proof of [8, Lemma 3.1], we have in σ ( J sG ) = (in σ ( J G )) s , for all s ≥ 1. Now, [4,Theorem 2.7] yields that in τ ( R ( J G )) = R (in σ ( J G )). By [11, Lemma 3.3], H is a bipartitegraph and every induced cycle in H has length 4. By [23, Theorem 1], F (in σ ( J G )) is aKoszul algebra. Therefore, the defining ideal of F (in σ ( J G )) is either zero ideal or a quadratichomogeneous ideal. It follows from [29, Theorem 3.1] that R (in σ ( J G )) is of fiber type. Thus,reltype(in σ ( J G )) ≤ 2, and hence, by [4, Corollary 2.8], reltype( J G ) ≤ 2. Since in τ ( R ( J G )) = R (in σ ( J G )) and { x i , y i : 1 ≤ i ≤ n } ∪ { in σ ( f e ) t : e ∈ E ( G ) } generate R (in σ ( J G )) as K -algebra, the set { x i , y i : 1 ≤ i ≤ n } ∪ { f e t : e ∈ E ( G ) } is a Sagbi basis of R ( J G ). (cid:3) We have seen in the above theorem that binomial edge ideals of closed graphs are ofquadratic type. Now, we characterize linear type closed graphs. Corollary 3.7. Let G be a closed graph on [ n ] with respect to given labelling of vertices.Then, J G is of linear type if and only if G is a K -free graph. ARVIND KUMAR Proof. Assume that J G is of linear type. By [20, Proposition 5.7], G is a K -free graph.Conversely, we assume that G is a K -free graph. Let H be the graph constructed in theproof of Theorem 3.3, i.e. I ( H ) = in σ ( J G ). Then, H is a forest. By [29, Corollary 3.2],in σ ( J G ) is of linear type, and hence, by [4, Corollary 2.8], J G is of linear type. (cid:3) Theorem 3.8. Let G be a closed graph on [ n ] . If char ( K ) = 0 , then R ( J G ) has rationalsingularities, and if char ( K ) > , then R ( J G ) is F -rational. In particular, R ( J G ) is aCohen-Macaulay normal domain.Proof. Assume, without loss of generality, that G is closed with respect to given labelling ofvertices. Let H be the graph constructed in the proof of Theorem 3.3, i.e. I ( H ) = in σ ( J G ).By [11, Lemma 3.3], H is a bipartite graph. It follows from [28, Corollary 5.3, Theorem5.9] that R (in σ ( J G )) is a Cohen-Macaulay normal domain. Now, Theorem 3.6 yields thatin τ ( R ( J G )) = R (in σ ( J G )). Thus, in τ ( R ( J G )) is a Cohen-Macaulay normal domain. Hence,the assertion follows from [4, Corollary 2.3]. (cid:3) Special fiber ring of binomial edge ideals of closed graphs In this section, we study the special fiber ring of binomial edge ideals of closed graphs.We begin with definitions. Let m denote the unique homogeneous maximal ideal of S . The special fiber ring of a homogeneous ideal I is the ring F ( I ) = R ( I ) / m R ( I ) ∼ = ⊕ k ≥ I k / m I k .The analytic spread of I is the Krull dimension of F ( I ), and it is denoted by ℓ ( I ). Theorem 4.1. Let G be a connected closed graph on [ n ] . Then,(1) F ( J G ) is a Koszul algebra.(2) if char ( K ) = 0 , then F ( J G ) has rational singularities.(3) if char ( K ) > , then F ( J G ) is F -rational.(4) F ( J G ) is a Cohen-Macaulay normal domain.(5) ℓ ( J G ) = 2 n − r − , where r is the number of indecomposable components of G .(6) ℓ ( J G ) = | E ( G ) | if and only if ω ( G ) ≤ .Proof. Assume, without loss of generality, that G is closed with respect to given labelling ofvertices. By [10, Theorem 2.10], { f e : e ∈ E ( G ) } is a Sagbi basis of the K -algebra F ( J G )with respect to the term order σ on S, i.e.,in σ ( F ( J G )) = F (in σ ( J G )) . (1) Let H be the graph constructed in the proof of Theorem 3.3, i.e. I ( H ) = in σ ( J G ). Itfollows from [11, Lemma 3.3] that H is a bipartite graph and every induced cycle in H haslength 4 . Thus, by [23, Theorem 1], F (in σ ( J G )) is a Koszul algebra. Hence, by [4, Corollary2.6], F ( J G ) is a Koszul algebra.(2-4) By [24, Corollary 1.3], F (in σ ( J G )) is normal. Now, the assertion follows from [4,Corollary 2.3].(5) By [4, Proposition 2.4], ℓ ( J G ) = ℓ (in σ ( J G )). Therefore, it is enough to find ℓ (in σ ( J G )).Let G , . . . , G r be the indecomposable components of G . Note that H = H ⊔ . . . ⊔ H r ,where H k is the connected bipartite graph such that I ( H k ) = in σ ( J G k ). Then, F ( I ( H )) = F ( I ( H )) ⊗ K · · ·⊗ K F ( I ( H r )). It follows from [29, Lemma 3.1, Proposition 3.2] that ℓ ( I ( H )) = r X i =1 | V ( H i ) | − r = r X i =1 (2 | V ( G i ) | − − r = 2 r X i =1 | V ( G i ) | − r = 2( n + r − − r = 2 n − r − . EES ALGEBRA AND SPECIAL FIBER RING OF BINOMIAL EDGE IDEALS OF CLOSED GRAPHS 7 (6) Note that H is a forest if and only if ω ( G ) ≤ 3. If ω ( G ) ≤ 3, then H is a forest,and therefore, ℓ ( J G ) = ℓ ( I ( H )) = | V ( H ) | − r = | E ( H ) | = | E ( G ) | . If ω ( G ) ≥ 4, then H isnot a forest, and therefore, ℓ ( J G ) = ℓ ( I ( H )) = | V ( H ) | − r < | E ( H ) | = | E ( G ) | . Hence, theassertion follows. (cid:3) As an immediate consequence, we obtain the following: Corollary 4.2. Let G = G ⊔ · · · ⊔ G c be a closed graph on [ n ] . Then,(1) F ( J G ) is a Koszul algebra.(2) if char ( K ) = 0 , then F ( J G ) has rational singularities.(3) if char ( K ) > , then F ( J G ) is F -rational.(4) F ( J G ) is a Cohen-Macaulay normal domain.(5) ℓ ( J G ) = 2 n − r − c , where r is the number of indecomposable components of G .(6) ℓ ( J G ) = | E ( G ) | if and only if ω ( G ) ≤ .Proof. The assertion follows from the fact that F ( J G ) = F ( J G ) ⊗ K · · · ⊗ K F ( J G c ), and byTheorem 4.1. (cid:3) We now obtain an upper bound for the regularity of special fiber ring of binomial edgeideals of closed graphs. Let { f , . . . , f m } be a minimal homogeneous generating set of anequi-generated homogeneous ideal I , and Q = K [ t , . . . , t m ] be a standard graded polynomialring over K . Let φ : Q → S be the K -algebra homomorphism given by φ ( t i ) = f i for all i .Then, Q/ ker( φ ) ≃ Im( φ ) = F ( I ), where ker( φ ) is a homogeneous ideal of Q and it is calledthe defining ideal of F ( I ). Notation 4.3. Let I be an equi-generated homogeneous ideal in S . Then, by reg( F ( I )) , wemean the regularity of Q/ ker( φ ) as Q -graded module. Theorem 4.4. Let G be a closed graph on [ n ] with respect to given labelling of vertices.Assume that G has no isolated vertices. Then, reg( F ( J G )) = reg( F (in σ ( J G ))) ≤ n − c ,where c is the number of components of G .Proof. First, we assume that G is a connected graph. It follows from the proof of Theorem4.1 that F ( J G ) and F (in σ ( J G )) are Cohen-Macaulay, and ℓ ( F ( J G )) = ℓ ( F (in σ ( J G ))). By[10, Theorem 2.10], in σ ( F ( J G )) = F (in σ ( J G )) . Now, it follows from [4, Proposition 2.4] thatHS F ( J G ) ( λ ) = HS in σ ( F ( J G )) ( λ ) = HS F (in σ ( J G )) ( λ ) , and therefore, h F ( J G ) ( λ ) = h F (in σ ( J G )) ( λ ) . Thus, reg( F ( J G )) = deg h F ( J G ) ( λ ) = deg h F (in σ ( J G )) ( λ ) = reg( F (in σ ( J G ))).Let H be the graph constructed in the proof of Theorem 3.3, i.e. I ( H ) = in σ ( J G ). Now,it follows from [12, Theorem 1] that reg( F ( I ( H ))) ≤ mat( H ) − n − 2, as {{ x i , y i +1 } :1 ≤ i ≤ n − } is a perfect matching of H . Hence, reg( F ( J G )) ≤ n − G = G ⊔ · · · ⊔ G c . Then, F ( J G ) = F ( J G ) ⊗ K · · · ⊗ K F ( J G c ), and F (in σ ( J G )) = F (in σ ( J G )) ⊗ K · · · ⊗ K F (in σ ( J G c )) which imples that reg( F ( J G )) = c X i =1 reg( F ( J G i )) = c X i =1 reg( F (in σ ( J G i ))) = reg( F (in σ ( J G ))) . Thus, reg( F ( J G )) = reg(in σ ( J G )) ≤ P ci =1 ( | V ( G i ) |− 2) = n − c. Hence, the assertion follows. (cid:3) ARVIND KUMAR As an immediate consequence, we obtain the following: Corollary 4.5. Let G be a closed graph on [ n ] . Assume that G has no isolated vertices.Then, reg( F ( J G )) ≤ n − c , where c is the number of connected components of G . Now, we obtain a lower bound for the regularity of special fiber ring of binomial edgeideals. Theorem 4.6. Let G be a graph on [ n ] . If H is an induced subgraph of G , then reg( F ( J H )) ≤ reg( F ( J G )) . In particular, reg( F ( J G )) ≥ ω ( G ) − if ω ( G ) ≥ .Proof. Let H be an induced subgraph of G . Clearly, F ( J H ) is a K -subalgebra of F ( J G ). Set R = K [ T { i,j } : { i, j } ∈ E ( H )] and R = K [ T { i,j } : { i, j } ∈ E ( G )]. Let I and I be idealsof R and R , respectively such that R /I ≃ F ( J H ), and R /I ≃ F ( J G ). Now, define π : R /I → R /I as π ( T { i,j } ) = 0 if { i, j } 6∈ E ( H ), and π ( T { i,j } ) = T { i,j } if { i, j } ∈ E ( H ).Consider, R /I i ֒ → R /I π −→ R /I . Then, π ◦ i is identity on R /I , and hence, R /I is analgebra retract of R /I . It follows from [22, Corollary 2.5] that reg( R /I ) ≤ reg( R /I ).Hence, reg( F ( J H )) ≤ reg( F ( J G )).Assume that ω ( G ) ≥ 4. Let H be a clique of G such that ω ( G ) = | V ( H ) | . Then, H is an induced subgraph of G . Since H is a complete graph, H is closed with respect tosome labelling of vertices. Assume, without loss of generality, that H is closed with respectto given labelling of vertices. By Theorem 4.4, reg( F ( J H )) = reg( F (in σ ( J H ))). Now, itfollows from [5, Proposition 5.7] that reg( F (in σ ( J H ))) = | V ( H ) | − ω ( G ) − 2. 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