Depth and Extremal Betti Number of Binomial Edge Ideals

TL;DR

This paper investigates the algebraic properties of binomial edge ideals of graphs, providing formulas for Betti numbers, depth, and Cohen-Macaulay defect, and constructing graphs with specific algebraic invariants.

Contribution

It offers explicit computations of Betti numbers for binomial edge ideals of cone graphs and constructs graphs with prescribed Cohen-Macaulay defect and extremal Betti numbers.

Findings

Betti numbers of $J_G$ expressed in terms of $J_H$

Depth of binomial edge ideals of join graphs determined

Existence of graphs with given regularity and extremal Betti number count

Abstract

Let $G$ be a simple graph on the vertex set $[n]$ and $J_G$ be the corresponding binomial edge ideal. Let $G=v*H$ be the cone of $v$ on $H$. In this article, we compute all the Betti numbers of $J_G$ in terms of Betti number of $J_H$ and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen-Macaulay defect of $S/J_G$ in terms of Cohen-Macaulay defect of $S_H/J_H$ and using this we construct a graph with Cohen-Macaulay defect $q$ for any $q\geq 1$. We obtain the depth of binomial edge ideal of join of graphs. Also, we prove that for any pair $(r,b)$ of positive integers with $1\leq b< r$, there exists a connected graph $G$ such that $reg(S/J_G)=r$ and the number of extremal Betti number of $S/J_G$ is $b$.

Equations

$$0⟶\underset{j\in \mathbb{Z}}{⨁}R(-p-j{)}^{{\beta }_{p,p+j}(M)}⟶\cdots ⟶\underset{j\in \mathbb{Z}}{⨁}R(-j{)}^{{\beta }_{0,j}(M)}⟶M⟶0,$$

$$\mathrm{reg}(M):=\max \{j:{\beta }_{i,i+j}^R(M)\ne 0\}$$

$${\mathrm{pd}}_R(M):=\max \{i:{\beta }_{i,i+j}^R(M)\ne 0\}.$$

$$0⟶\frac{S}{J_G}⟶\frac{S}{(x_v,y_v)+J_H}\oplus \frac{S}{J_{K_{n+1}}}⟶\frac{S}{(x_v,y_v)+J_{K_n}}⟶0.$$

$${\beta }_{i,i+j}^T\left(\frac{T}{I+J}\right)=\sum _{\begin{array}{c}{i}_1+i_2=i\\ j_1+j_2=j\end{array}}{\beta }_{i_1,i_1+j_1}^R\left(\frac{R}{I}\right){\beta }_{i_2,i_2+j_2}^{R^{\prime }}\left(\frac{R^{\prime }}{J}\right).$$

$$0\to {\mathrm{Tor}}_{n+1,n+1+j}^S\left(\frac{S}{J_G},\mathbb{K}\right)\to {\mathrm{Tor}}_{n+1,n+1+j}^S\left(\frac{S}{(x_v,y_v)+J_H},\mathbb{K}\right)\to 0.$$

$$0p{t}_S(S/J_G)\ge \min \{0p{t}_{S_H}(S_H/J_H),n+2\}=n+1.$$

$$0p{t}_S(S/J_G)=\min \{0p{t}_{S_H}(S_H/J_H),n+2\}.$$

$$0\to {\mathrm{Tor}}_{n+1,n+1+j}^S\left(\frac{S}{J_G},\mathbb{K}\right)\to {\mathrm{Tor}}_{n+1,n+1+j}^S\left(\frac{S}{(x_v,y_v)+J_H},\mathbb{K}\right)\to 0.$$

$$\beta_{i,i+j}^{S}\left(\frac{S}{J_{G}}\right)=\left\{\begin{array}[]{ll}i\left({\sf k}_{i}(H)+{\sf k}_{i+1}(H)\right),&\text{ if }j=1\\ \beta_{i,i+2}^{S_{H}}\left(\frac{S_{H}}{J_{H}}\right)+2\beta_{i-1,i+1}^{S_{H}}\left(\frac{S_{H}}{J_{H}}\right)+\beta_{i-2,i}^{S_{H}}\left(\frac{S_{H}}{J_{H}}\right)\\ +(i-1)\binom{n+1}{i+1}-(i-1){\sf k}_{i}(H)-(i-1){\sf k}_{i+1}(H),&\text{ if }j=2\\ \beta_{i,i+j}^{S_{H}}\left(\frac{S_{H}}{J_{H}}\right)+2\beta_{i-1,i-1+j}^{S_{H}}\left(\frac{S_{H}}{J_{H}}\right)+\beta_{i-2,i-2+j}^{S_{H}}\left(\frac{S_{H}}{J_{H}}\right),&\text{ if }j\geq 3,\end{array}\right.$$

$$\begin{gathered} \cdots \to {\mathrm{Tor}}_{i,i+j}^S\left(\frac{S}{J_G},\mathbb{K}\right)\to {\mathrm{Tor}}_{i,i+j}^S\left(\frac{S}{(x_v,y_v)+J_H},\mathbb{K}\right)\oplus {\mathrm{Tor}}_{i,i+j}^S\left(\frac{S}{J_{K_{n+1}}},\mathbb{K}\right) \\ \to {\mathrm{Tor}}_{i,i+j}^S\left(\frac{S}{(x_v,y_v)+J_{K_n}},\mathbb{K}\right)\to {\mathrm{Tor}}_{i-1,i+j}^S\left(\frac{S}{J_G},\mathbb{K}\right)\to \cdots \end{gathered}$$

$${\beta }_{i,i+2}^S\left(\frac{S}{J_G}\right)$$

$$-{\beta }_{i+1,i+2}^S\left(\frac{S}{(x_v,y_v)+J_H}\right)+{\beta }_{i+1,i+2}^S\left(\frac{S}{J_G}\right).$$

$${\beta }_{i+1,i+2}^S\left(\frac{S}{((x_v,y_v)+J_{K_n})}\right)$$

$$=(i+1)\left(\genfrac{}{}{0ex}{}{n}{i+2}\right)+2i\left(\genfrac{}{}{0ex}{}{n}{i+1}\right)+(i-1)\left(\genfrac{}{}{0ex}{}{n}{i}\right)$$

$${\beta }_{i,i+2}^S\left(\frac{S}{(x_v,y_v)+J_H}\right)={\beta }_{i,i+2}^{S_H}\left(\frac{S_H}{J_H}\right)+2{\beta }_{i-1,i+1}^{S_H}\left(\frac{S_H}{J_H}\right)+{\beta }_{i-2,i}^{S_H}\left(\frac{S_H}{J_H}\right).$$

$${\beta }_{i,i+2}^S\left(\frac{S}{J_G}\right)$$

$$+(i-1)\left(\genfrac{}{}{0ex}{}{n+1}{i+1}\right)-(i-1){\mathsf{k}}_i(H)-(i-1){\mathsf{k}}_{i+1}(H).$$

$${\mathrm{Tor}}_{i,i+j}^S\left(\frac{S}{J_{K_{n+1}}},\mathbb{K}\right)={\mathrm{Tor}}_{i,i+j}^S\left(\frac{S}{(x_v,y_v)+J_{K_n}},\mathbb{K}\right)={\mathrm{Tor}}_{i+1,i+j}^S\left(\frac{S}{(x_v,y_v)+J_{K_n}},\mathbb{K}\right)=0.$$

$${\beta }_{i,i+j}^S\left(\frac{S}{J_G}\right)={\beta }_{i,i+j}^{S_H}\left(\frac{S_H}{J_H}\right)+2{\beta }_{i-1,i-1+j}^{S_H}\left(\frac{S_H}{J_H}\right)+{\beta }_{i-2,i-2+j}^{S_H}\left(\frac{S_H}{J_H}\right),$$

$$\begin{array}[]{c|ccccccccccc}&0&1&2&\cdots&i&i+1&i+2&\cdots&n&n+1&n+2\\ \hline\cr 0&1&0&0&\cdots&0&0&0&\cdots&0&0&0\\ 1&0&\beta_{1,2}&\beta_{2,3}&\cdots&0&0&0&\cdots&0&0&0\\ 2&0&0&\beta_{2,4}&\cdots&\beta_{i,i+2}&\beta_{i+1,i+3}&\beta_{i+2,i+4}&\cdots&\beta_{n,n+2}&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\vdots\\ i&0&0&0&\cdots&\beta_{i,2i}&\beta_{i+1,2i+1}&\beta_{i+2,2i+2}&\ddots&\ddots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\vdots\\ n-2&0&0&\beta_{2,n}&\cdots&\beta_{i,n-2+i}&\beta_{i+1,n-1+i}&\beta_{i+2,n+i}&\cdots&\beta_{n,2n-2}&\beta_{n+1,2n-1}&\beta_{n+2,2n}\par\end{array}$$

$${\beta }_{p+2,p+4}^S(S/J_G)={\beta }_{p,p+2}^{S_H}(S_H/J_H)+(p+1)\left(\genfrac{}{}{0ex}{}{n+1}{p+3}\right)-(p+1){\mathsf{k}}_{p+2}(H)-(p+1){\mathsf{k}}_{p+3}(H).$$

$${\beta }_{k,k+2}^S\left(\frac{S}{J_G}\right)={\beta }_{k-2,k}^{S_H}\left(\frac{S_H}{J_H}\right)+(k-1)\left(\genfrac{}{}{0ex}{}{n+1}{k+1}\right)-(k-1)({\mathsf{k}}_k(H)+{\mathsf{k}}_{k+1}(H))={\beta }_{k-2,k}^{S_H}\left(\frac{S_H}{J_H}\right).$$

$$J_{G}=P_{\emptyset}(G)\cap\big{(}(x_{i},y_{i}:i\in[n_{1}])+J_{G_{2}}\big{)}\cap\big{(}(x_{j},y_{j}:j\in[n_{2}])+J_{G_{1}}\big{)}.$$

$$0⟶\frac{S}{Q_3}⟶\frac{S}{P_{\varnothing }(G)}\oplus \frac{S}{Q_2}⟶\frac{S}{(x_i,y_i:i\in [n_1])+J_{K_{n_2}}}⟶0$$

$$0⟶\frac{S}{J_G}⟶\frac{S}{Q_1}\oplus \frac{S}{Q_3}⟶\frac{S}{(x_i,y_i:i\in [n_2])+J_{K_{n_1}}}⟶0.$$

$$0p{t}_S(S/Q_3)\ge \min \{0p{t}_{S_2}(S_2/J_{G_2}),n_2+2\}$$

$$0p{t}_S(S/J_G)\ge \min \{0p{t}_{S_1}(S_1/J_{G_1}),0p{t}_{S_2}(S_2/J_{G_2}),n_1+2,n_2+2\}.$$

$$0p{t}_S(S/J_G)=\underset{i=1,2}{\min }0p{t}_{S_i}(S_i/J_{G_i}).$$

$$0\to {\mathrm{Tor}}_{p,p+j}^S\left(\frac{S}{Q_3},\mathbb{K}\right)\to {\mathrm{Tor}}_{p,p+j}^S\left(\frac{S}{Q_2},\mathbb{K}\right)\to {\mathrm{Tor}}_{p,p+j}^S\left(\frac{S}{P_{\varnothing }(G)+Q_2},\mathbb{K}\right)\to \cdots$$

$$0\to {\mathrm{Tor}}_{q,q+l}^S\left(\frac{S}{J_G},\mathbb{K}\right)\to {\mathrm{Tor}}_{q,q+l}^S\left(\frac{S}{Q_1},\mathbb{K}\right)\oplus {\mathrm{Tor}}_{q,q+l}^S\left(\frac{S}{Q_3},\mathbb{K}\right)\to 0.$$

$$0p{t}_S(S/J_G)=\min \{0p{t}_{S_1}(S_1/J_{G_1}),0p{t}_{S_2}(S_2/J_{G_2}),n_2+2\}.$$

Full text

AMS Subject Classification (2010): 13D02,13C13, 05E40

1. Introduction

Let $R=\mathbb{K}[x_{1},\ldots,x_{m}]$ be the polynomial ring over an arbitrary field $\mathbb{K}$ and $M$ be a finitely generated graded $R$-module. Let

[TABLE]

be the minimal graded free resolution of $M$, where $R(-j)$ is the free $R$-module of rank $1$ generated in degree $j$. The number $\beta_{i,i+j}^{R}(M)$ is the $(i,i+j)$-th graded Betti number of $M$. The projective dimension and Castelnuovo-Mumford regularity are two invariants associated with $M$ that can be read off from the minimal free resolution. The Castelnuovo-Mumford regularity of $M$, denoted by $\operatorname{reg}(M)$, is defined as

[TABLE]

and the projective dimension of $M$, denoted by $\operatorname{pd}_{R}(M)$, is defined as

[TABLE]

A nonzero graded Betti number $\beta_{i,i+j}^{R}(M)$ is called an extremal Betti number, if $\beta_{r,r+s}^{R}(M)=0$ for all pairs $(r,s)\neq(i,j)$ with $r\geq i$ and $s\geq j$. Observe that the extremal Betti number is unique if and only if $\beta_{p,p+r}^{R}(M)\neq 0$, where $p=\operatorname{pd}_{R}(M)$ and $r=\operatorname{reg}(M)$.

Let $G$ be a simple graph on $V(G)=\{1,2,\dots,n\}$ and edge set $E(G)$. Let $S=\mathbb{K}[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]$ be the polynomial ring over an arbitrary field $\mathbb{K}$. The ideal $J_{G}$ generated by the binomials $x_{i}y_{j}-x_{j}y_{i}$, where $i<j$ and $\{i,j\}\in E(G)$, is known as the binomial edge ideal of $G$. The notion of binomial edge ideal was introduced by Herzog et.al. in [5] and independently by Ohtani in [19]. Algebraic properties and invariants of binomial edge ideals have been studied by many authors, see [4, 16, 21]. In particular, establishing a relationship between Castelnuovo-Mumford regularity (simply regularity), projective dimension, Hilbert series of binomial edge ideals and combinatorial invariants associated with graphs is an active area of research, see [1, 11, 13, 18, 20]. In general, the algebraic invariants such as regularity and depth of $J_{G}$ are hard to compute. There are bounds known for the regularity and depth of binomial edge ideals, see [1, 17]. The maximal possible depth of binomial edge ideal of a connected graph on $n$ vertices is $n+1$ (see [1, Theorems 3.19, 3.20]). Also, if $G$ is a connected graph on $n$ vertices such that $S/J_{G}$ is Cohen-Macaulay, then $0pt_{S}(S/J_{G})=n+1$. In [2], de Alba and Hoang studied the depth of some subclass of closed graphs. However not much more is known about the depth of binomial edge ideal. For an ideal $I\subset S$, the Cohen-Macaulay defect of $S/I$ is defined to be $\operatorname{cmdef}(S/I):=\dim(S/I)-0pt_{S}(S/I)$. We study the depth and Cohen-Macaulay defect of $S/J_{G}$, where $G$ is a cone of $v$ on a graph $H$, denoted by $v*H$ (for definition see section 3). We show that the depth remains invariant under the process of taking cone on connected graph, (Theorem 3.4). As a consequence, we prove that for any positive integer $q$, there exists a graph having Cohen-Macaulay defect equal to $q$, (Corollary 3.6). We also compute the depth of $S/J_{v*H}$, when $H$ is a disconnected graph, (Theorem 3.9).

Another homological invariant that helps in understanding more about its structure is the Betti number. There have been few attempts in computing the Betti numbers of binomial edge ideals, for example, Zafar and Zahid for cycles, [24], Schenzel and Zafar for complete bipartite graphs, [23], Jayanthan et al. for trees and unicyclic graphs [12]. Extremal Betti numbers of binomial edge ideals of closed graphs were studied by de Alba and Hoang in [2]. In [7], Herzog and Rinaldo studied extremal Betti number of binomial edge ideal of block graphs. We compute all the Betti numbers of cone of a graph, (Theorem 3.10). As a consequence, we obtain the Betti numbers of binomial edge ideal of wheel graph, (Corollary 3.11).

We then consider a more general form of cone, namely the join product of two arbitrary graphs. Given two graphs $G_{1}$ and $G_{2}$, it is interesting to understand the properties of $G_{1}*G_{2}$ (for definition see section 4) in terms of the corresponding properties of $G_{1}$ and $G_{2}$. In [22], Kiani and Saeedi Madani studied the regularity of $J_{G_{1}*G_{2}}$. We computed the Hilbert series of binomial edge ideal of $G_{1}*G_{2}$ in terms of the Hilbert series of $J_{G_{1}}$ and $J_{G_{2}}$, [15]. In this article, we study the depth of $S/J_{G_{1}*G_{2}}$, (Theorems 4.1, 4.3, 4.4). As a consequence, we obtain the depth of complete multipartite graphs, (Corollary 4.5).

Recently, researchers are trying to construct graphs such that their corresponding edge ideals satisfy certain algebraic properties. For a given pair of positive integers $(r,s)$, Hibi and Matsuda in [10] showed the existence of monomial ideal $I_{r,s}$ such that $\operatorname{reg}(S/I_{r,s})=r$ and the degree of $h$-polynomial of $S/I_{r,s}$ is $s$. In [8], Hibi et al. constructed a graph $G$ such that for $1\leq b\leq r$, the regularity of the monomial edge ideal of $G$ is $r$ and the number of its extremal Betti numbers is $b$. Given a pair $(r,s)$ with $1\leq r\leq s$, Hibi and Matsuda constructed a graph $G$ such that $\operatorname{reg}(S/J_{G})=r$ and the degree of $h$-polynomial of $S/J_{G}$ is $s$, [9]. In this article, we construct a graph $G$ such that $\operatorname{reg}(S/J_{G})=r$ and the number of extremal Betti numbers of $S/J_{G}$ is $b$, for $1\leq b<r$ (Theorem 5.4).

2. Preliminaries

In this section, we recall some notation and fundamental results on graphs and their corresponding binomial edge ideals.

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. For $A\subseteq V(G)$, $G[A]$ denotes the induced subgraph of $G$ on the vertex set $A$, that is, for $i,j\in A$, $\{i,j\}\in E(G[A])$ if and only if $\{i,j\}\in E(G)$. For a vertex $v$, $G\setminus v$ denotes the induced subgraph of $G$ on the vertex set $V(G)\setminus\{v\}$. A vertex $v\in V(G)$ is said to be a cut vertex if $G\setminus v$ has more components than $G$. We say that $G$ is $k$ vertex-connected if $k<n$ and for every $A\subset[n]$ with $|A|<k$, the induced graph $G[\bar{A}]$ is connected, where $\bar{A}=[n]\setminus A$. The vertex connectivity of a connected graph $G$, denoted by $\kappa(G)$, is defined as the maximum positive integer $k$ such that $G$ is $k$ vertex-connected. A subset $U$ of $V(G)$ is said to be a clique if $G[U]$ is a complete graph. We denote the number of cliques of cardinality $i$ in $G$ by ${\sf k}_{i}(G)$. A vertex $v$ is said to be a simplicial vertex if it belongs to exactly one maximal clique. For a vertex $v$, $N_{G}(v)=\{u\in V(G)\leavevmode\nobreak\ :\leavevmode\nobreak\ \{u,v\}\in E(G)\}$ denotes neighborhood of $v$ and $G_{v}$ is the graph on the vertex set $V(G)$ and edge set $E(G_{v})=E(G)\cup\{\{u,w\}:u,w\in N_{G}(v)\}$. A component of $G$ is said to be a nontrivial component if it has atleast one edge.

For $T\subset[n]$, let $\bar{T}=[n]\setminus T$ and $c_{T}$ denote the number of connected components of $G[\bar{T}]$. Let $G_{1},\cdots,G_{c_{T}}$ be the connected components of $G[\bar{T}]$. For each $i$, let $\tilde{G_{i}}$ denote the complete graph on $V(G_{i})$ and $P_{T}(G)=(\underset{i\in T}{\cup}\{x_{i},y_{i}\},J_{\tilde{G_{1}}},\cdots,J_{\tilde{G}_{c_{T}}}).$ It was shown by Herzog et al. that $J_{G}=\underset{T\subseteq[n]}{\cap}P_{T}(G)$, [5, Theorem 3.2]. For each $i\in T$, if $i$ is a cut vertex of the graph $G[\bar{T}\cup\{i\}]$, then we say that $T$ has the cut point property. Set $\mathcal{C}(G)=\{\emptyset\}\cup\{T:T\;\text{has the cut point property}\}$. It follows from [5, Corollary 3.9] that $T\in\mathcal{C}(G)$ if and only if $P_{T}(G)$ is a minimal prime of $J_{G}$. It follows from the Auslander-Buchsbaum formula that $0pt_{S}(S/J_{G})=2n-\operatorname{pd}_{S}(S/J_{G})$.

The following basic property of depth is used repeatedly in this article.

Lemma 2.1**.**

Let $S$ be a standard graded polynomial ring and $M,N$ and $P$ be finitely generated graded $S$-modules. If $0\rightarrow M\xrightarrow{f}N\xrightarrow{g}P\rightarrow 0$ is a short exact sequence with $f,g$ graded homomorphisms of degree zero, then

1. (i)

$0pt_{S}(M)\geq\min\{0pt_{S}(N),0pt_{S}(P)+1\},$** 2. (ii)

$0pt_{S}(M)=0pt_{S}(P)+1$* if $0pt_{S}(N)>0pt_{S}(P)$,* 3. (iii)

$0pt_{S}(M)=0pt_{S}(N)$* if $0pt_{S}(N)<0pt_{S}(P)$.*

3. Binomial edge ideal of cone of a graph

In this section, we study the binomial edge ideal of cone of a graph. Let $H$ be a graph on the vertex set $[n]$. The cone of $v$ on $H$, denoted by $v*H$, is the graph with the vertex set $V(v*H)=V(H)\sqcup\{v\}$ and edge set $E(v*H)=E(H)\sqcup\{\{v,u\}\mid u\in V(H)\}.$ From now, we assume that $H$ is not a complete graph. Set $G=v*H$, $S_{H}=\mathbb{K}[x_{i},y_{i}:i\in V(H)]$ and $S=S_{H}[x_{v},y_{v}]$. First, we recall a lemma due to Ohtani which is useful in this section.

Lemma 3.1**.**

([19, Lemma 4.8]) Let $G$ be a graph on $V(G)$ and $v\in V(G)$ such that $v$ is not a simplicial vertex. Then $J_{G}=(J_{G\setminus v}+(x_{v},y_{v}))\cap J_{G_{v}}$.

One can see that if $G=v*H$, then ${G_{v}}=K_{n+1}$, $G_{v}\setminus v=K_{n}$ and $G\setminus v=H$. Therefore, $(x_{v},y_{v})+J_{G\setminus v}+J_{G_{v}}=(x_{v},y_{v})+J_{K_{n}}$. Thus, by Lemma 3.1, we have the following short exact sequence:

[TABLE]

Remark 3.2**.**

It follows from [3, Theorem 1.1] that if $G$ is a complete graph on $[n]$, then $S/J_{G}$ is Cohen-Macaulay of dimension $n+1$. If $G$ is a connected graph which is not a complete graph, then $\kappa(G)\geq 1$. Therefore, by [1, Theorems 3.19, 3.20], we get $\operatorname{pd}_{S}(S/J_{G})\geq n-2+\kappa(G)\geq n-1$. Thus, for any connected graph $G$, $\operatorname{pd}_{S}(S/J_{G})\geq n-1$ and hence, by Auslander-Buchsbaum formula, $0pt_{S}(S/J_{G})\leq n+1$.

We proceed to prove the following lemma which plays an important role.

Lemma 3.3**.**

Let $G$ be a connected graph on the vertex set $[n]$. Let $p=\operatorname{pd}_{S}(S/J_{G})$. Then $\beta_{p,p+1}^{S}(S/J_{G})\neq 0$ if and only if $G$ is a complete graph. Moreover, if $\beta_{i,i+2}^{S}(S/J_{G})$ is an extremal Betti number, then $i=p$.

Proof.

By Remark 3.2, $p\geq n-1$. It follows from [6, Corollary 4.3] that $\beta_{p,p+1}^{S}(S/J_{G})=p{\sf k}_{p+1}(G)$. Therefore, $\beta_{p,p+1}^{S}(S/J_{G})\neq 0$ if and only if $G$ is a complete graph. Now, if possible assume that $i<p$. Since $\beta_{i,i+2}^{S}(S/J_{G})$ is an extremal Betti number, $\beta_{p,p+j}^{S}(S/J_{G})=0$ for $j\geq 2$, which implies that $\beta_{p,p+1}^{S}(S/J_{G})$ must be an extremal Betti number. Thus, $G$ is a complete graph which contradicts [21, Theorem 2.1], as $\operatorname{reg}(S/J_{G})\geq 2$. Hence, the assertion follows. ∎

Let $M$ be a finite graded $S$-module. The Cohen-Macaulay defect, denoted by $\operatorname{cmdef}(M)$, is defined by $\dim(M)-0pt_{S}(M)$. A graded $S$-module $M$ is said to be almost Cohen-Macaulay if $\operatorname{cmdef}(M)=1$. A graph $G$ is said to be (almost) Cohen-Macaulay if $S/J_{G}$ is (almost) Cohen-Macaulay.

First, we recall some basic facts about Betti numbers and minimal free resolution. Let $R=\mathbb{K}[x_{1},\dots,x_{m}]$, $R^{\prime}=\mathbb{K}[x_{m+1},\dots,x_{n}]$ and $T=\mathbb{K}[x_{1},\dots,x_{n}]$ be polynomial rings. Let $I\subseteq R$ and $J\subseteq R^{\prime}$ be homogeneous ideals. Then minimal free resolution of $T/(I+J)$ is tensor product of minimal free resolutions of $R/I$ and $R^{\prime}/J$. Also, for all $i,j$,

[TABLE]

Now, we construct almost Cohen-Macaulay graphs.

Theorem 3.4**.**

Let $H$ be a connected graph on the vertex set $[n]$ and $G=v*H$ be the cone of $v$ on $H$. Then, $0pt_{S}(S/J_{G})=0pt_{S_{H}}(S_{H}/J_{H}).$ In particular, if $H$ is Cohen-Macaulay, then $G$ is almost Cohen-Macaulay.

Proof.

Assume that $0pt_{S_{H}}(S_{H}/J_{H})=n+1$. Therefore, $\operatorname{pd}_{S_{H}}(S_{H}/J_{H})=n-1$ and $\operatorname{pd}_{S}(S/((x_{v},y_{v})+J_{H}))=n+1$. Also, we have $\operatorname{pd}_{S}(S/((x_{v},y_{v})+J_{K_{n}}))=n+1$ and $\operatorname{pd}_{S}(S/J_{K_{n+1}})=n$. Since, $H$ is a connected graph, by Lemma 3.3, there exists a $j\geq 2$ such that $\beta_{n-1,n-1+j}^{S_{H}}(S_{H}/J_{H})\neq 0$. Consider, the long exact sequence of Tor corresponding to (1),

[TABLE]

Since $\beta_{n+1,n+1+j}^{S}(S/((x_{v},y_{v})+J_{H}))\neq 0$, $\beta_{n+1,n+1+j}^{S}(S/J_{G})\neq 0$. Therefore, $\operatorname{pd}_{S}(S/J_{G})\geq n+1$ and hence by Auslander-Buchsbaum formula, $0pt_{S}(S/J_{G})\leq n+1$. Now using Lemma 2.1 on the short exact sequence (1), we get that

[TABLE]

Hence, $0pt_{S}(S/J_{G})=n+1$. If $0pt_{S_{H}}(S_{H}/J_{H})<n+1$, then by Lemma 2.1, $0pt_{S}(S/J_{G})=0pt_{S_{H}}(S_{H}/J_{H}).$ Now, if $H$ is Cohen-Macaulay, then $0pt_{S}(S/J_{G})=n+1$. It follows from [15, Theorem 4.6] that $\dim(S/J_{G})=n+2$. Hence, $G$ is an almost Cohen-Macaulay. ∎

Theorem 3.5**.**

Let $H$ be a connected graph on the vertex set $[n]$ and $G=v*H$ be the cone of $v$ on $H$. If $\dim(S_{H}/J_{H})\geq n+2$, then $\operatorname{cmdef}(S/J_{G})=\operatorname{cmdef}(S_{H}/J_{H})$ and otherwise $\operatorname{cmdef}(S/J_{G})=\operatorname{cmdef}(S_{H}/J_{H})+1.$

Proof.

It follows from [15, Theorem 4.6] that if $\dim(S_{H}/J_{H})\geq n+2$, then $\dim(S/J_{G})=\dim(S_{H}/J_{H})$. Thus, by Theorem 3.4, $\operatorname{cmdef}(S/J_{G})=\operatorname{cmdef}(S_{H}/J_{H})$. Now, if $\dim(S_{H}/J_{H})=n+1$, then again by [15, Theorem 4.6], $\dim(S/J_{G})=n+2$ and hence $\operatorname{cmdef}(S/J_{G})=\operatorname{cmdef}(S_{H}/J_{H})+1.$ ∎

We now show that one can construct graphs with as large Cohen-Macaulay defect as one wants.

Corollary 3.6**.**

Let $H$ be a connected graph on $[n]$ and $q$ be a positive integer. If $G=K_{q}*H$, then $0pt_{S}(S/J_{G})=0pt_{S_{H}}(S_{H}/J_{H})$. In particular, if $H$ is Cohen-Macaulay, then $\operatorname{cmdef}(S/J_{G})=q$.

Proof.

Let $v_{1},\ldots,v_{q}$ be vertices of $K_{q}$. Observe that $K_{q}*H=v_{1}*(\cdots*(v_{q}*H)\cdots)$. By recursively applying Theorem 3.4, $0pt_{S}(S/J_{G})=0pt_{S_{H}}(S_{H}/J_{H})$. Now, if $H$ is Cohen-Macaulay, then $0pt_{S}(S/J_{G})=n+1$ and it follows from [15, Theorem 4.12] that $\dim(S/J_{G})=n+q+1$. Hence, the assertion follows. ∎

Let $G=K_{q}*H$, then by [15, Theorem 4.12] and Corollary 3.6, if $\dim(S_{H}/J_{H})\geq n+q+1$, then $\operatorname{cmdef}(S/J_{G})=\operatorname{cmdef}(S_{H}/J_{H})$ otherwise $\operatorname{cmdef}(S/J_{G})=n+q+1-\dim(S_{H}/J_{H})+\operatorname{cmdef}(S_{H}/J_{H})$.

To compute the depth formula for cone of a disconnected graph, we need the following lemma.

Lemma 3.7**.**

Let $G$ be a disconnected graph on the vertex set $[n]$. Assume that $G$ has atleast two nontrivial components. Let $p=\operatorname{pd}_{S}(S/J_{G})$. Then $\beta_{p,p+1}^{S}(S/J_{G})=0$. Moreover, if $\beta_{i,i+2}^{S}(S/J_{G})$ is an extremal Betti number, then $i=p$.

Proof.

Let $H_{1},\ldots,H_{q}$ be nontrivial connected components of $G$ with $q\geq 2$. By Remark 3.2, for each $i\in[q]$, $\operatorname{pd}_{S_{H_{i}}}(S_{H_{i}}/J_{H_{i}})\geq|V(H_{i})|-1$, where $S_{H_{i}}=\mathbb{K}[x_{v},y_{v}:v\in V(H_{i})]$. Let $m=\sum_{i=1}^{q}|V(H_{i})|$. Thus, $p\geq m-q.$ It follows from [6, Corollary 4.3], that $\beta_{p,p+1}^{S}(S/J_{G})=p{\sf k}_{p+1}(G)$. If possible, $\beta_{p,p+1}^{S}(S/J_{G})\neq 0$, then $G$ has an induced clique of size atleast $m-q+1$, which is a contradiction. Now, if possible assume that $i<p$ which implies that $\beta_{p,p+1}^{S}(S/J_{G})$ is an extremal Betti number, which is a contradiction as $\beta_{p,p+1}^{S}(S/J_{G})=0$. Hence, the assertion follows. ∎

Remark 3.8**.**

Let $G$ be a disconnected graph on $[n]$. If $0pt_{S}(S/J_{G})=n+1$, then either $G$ has atleast two nontrivial components or $G$ has exactly one nontrivial component which is not a complete graph. Moreover, $\beta_{n-1,n}^{S}(S/J_{G})=0$.

We now compute the depth formula for cone of a disconnected graph.

Theorem 3.9**.**

Let $G=v*H$, where $H$ is a disconnected graph on $[n]$. Then

[TABLE]

Proof.

If $0pt_{S_{H}}(S/J_{H})<n+1$, then by using Lemma 2.1 in the short exact sequence (1), we have $0pt_{S}(S/J_{G})=0pt_{S_{H}}(S_{H}/J_{H}).$ Also, if $0pt_{S_{H}}(S_{H}/J_{H})>n+1$, then by virtue of Lemma 2.1 $0pt_{S}(S/J_{G})=n+2$. Now, assume that $0pt_{S_{H}}(S_{H}/J_{H})=n+1$. Observe that $\operatorname{pd}_{S_{H}}(S_{H}/J_{H})=n-1$, $\operatorname{pd}_{S}(S/((x_{v},y_{v})+J_{H}))=n+1$ and $\operatorname{pd}_{S}(S/((x_{v},y_{v})+J_{K_{n}}))=n+1$. By Remark 3.8, there exists $j\geq 2$ such that $\beta_{n-1,n-1+j}^{S_{H}}(S_{H}/J_{H})\neq 0$. Now consider, the long exact sequence of Tor corresponding to (1),

[TABLE]

Since, $\beta_{n+1,n+1+j}^{S}(S/((x_{v},y_{v})+J_{H}))\neq 0$ and hence $\beta_{n+1,n+1+j}^{S}(S/J_{G})\neq 0$. Therefore, $\operatorname{pd}_{S}(S/J_{G})\geq n+1$ and hence $0pt_{S}(S/J_{G})\leq n+1$. Using Lemma 2.1, we have $0pt_{S}(S/J_{G})\geq n+1$ and this completes the proof. ∎

Also, if $G=K_{q}*H$, where $H$ is a disconnected graph, then by Theorems 3.4, 3.9, $0pt_{S}(S/J_{G})=\min\{0pt_{S_{H}}(S_{H}/J_{H}),n+2\}$. Now we compute the Betti numbers of $S/J_{v*H}$ in terms of the Betti numbers of $S_{H}/J_{H}$.

Theorem 3.10**.**

Let $H$ be a graph on the vertex set $[n]$. Let $G=v*H$ be the cone of $v$ on $H$. Then, for $i,j$,

[TABLE]

where $\beta_{i-2,i-2+j}^{S}\left(\frac{S}{J_{G}}\right)=0$ and $\beta_{i-1,i-1+j}^{S}\left(\frac{S}{J_{G}}\right)=0$, if $i-2<0$ and $i-1<0$ respectively.

Proof.

It follows from [6, Corollary 4.3] that $\beta_{i,i+1}^{S}(S/J_{G})=i{\sf k}_{i+1}(G).$ Let $U$ be a clique in $G$ on $(i+1)$-vertices. Then either $v\in U$ or $v\notin U$. If $v\notin U$, then $U$ is a clique in $H$ on $(i+1)$-vertices. If $v\in U$, then $U\setminus\{v\}$ is a clique in $H$ on $i$-vertices. Therefore, ${\sf k}_{i+1}(G)={\sf k}_{i}(H)+{\sf k}_{i+1}(H)$ and hence $\beta_{i,i+1}^{S}(S/J_{G})=i\left({\sf k}_{i}(H)+{\sf k}_{i+1}(H)\right)$. Now, consider the long exact sequence of Tor modules corresponding to the short exact sequence (1):

[TABLE]

For $j=2$, the above long exact sequence of Tor gives us

[TABLE]

By (2), we have

[TABLE]

and

[TABLE]

Therefore, we have

[TABLE]

Now let $j\geq 3$. Since, $\operatorname{reg}(S/((x_{v},y_{v})+J_{K_{n}}))=\operatorname{reg}(S/J_{K_{n+1}})=1$,

[TABLE]

Then for $j\geq 3$, $\operatorname{Tor}_{i,i+j}^{S}\left(\frac{S}{J_{G}},\mathbb{K}\right)\simeq\operatorname{Tor}_{i,i+j}^{S}\left(\frac{S}{(x_{v},y_{v})+J_{H}},\mathbb{K}\right)$ and hence by virtue of (2), we have

[TABLE]

which proves our result. ∎

Let $G=K_{q}*H$ be the join of a complete graph and $H$. Then by using the above theorem recursively, one can compute all the Betti numbers of $S/J_{G}$. Now, we compute the Betti diagram of the wheel graph. The wheel graph, denoted by $W_{n}$, is the cone of $v$ on $C_{n}$, $n\geq 4$.

Corollary 3.11**.**

Let $W_{n}=v*C_{n}$ be the wheel graph with $n\geq 4$. Then $\operatorname{reg}(S/J_{W_{n}})=n-2$, $\operatorname{pd}_{S}(S/J_{W_{n}})=n+2$ and the Betti diagram of $S/J_{W_{n}}$ looks like the following:

[TABLE]

where, $\beta_{1,2}=2n$, $\beta_{2,3}=2n$, $\beta_{2,4}=\binom{n}{2}+\binom{n+1}{3}-n$, $\beta_{3,5}=2\binom{n}{2}+2\binom{n+1}{4}$, $\beta_{4,6}=\binom{n}{2}+3\binom{n+1}{5}$,

$\beta_{i,i+2}=(i-1)\binom{n+1}{i+1}\text{, if }i=5,\ldots,n,$**

$\beta_{i,2i}=\beta_{i+2,2i+2}=\binom{n}{i},\beta_{i+1,2i+1}=2\binom{n}{i}\text{, if }i=3,\ldots,n-3,$**

$\beta_{i,i+n-2}=(n+1-i)\binom{n+2}{i-2}+2\binom{n+1}{i-3}\text{, if }i=2,\ldots,n-3,$**

$\beta_{n-2,2n-4}=\binom{n}{2}+3\binom{n+2}{6}+2\binom{n+1}{6}$, $\beta_{n-1,2n-3}=2\binom{n+1}{3}+2\binom{n}{4}+4\binom{n+1}{5}$, $\beta_{n,2n-2}=\binom{n-1}{2}-1+\binom{n+2}{4}+2\binom{n+1}{4}$, $\beta_{n+1,2n-1}=2\binom{n-1}{2}-2+2\binom{n}{3}$ and $\beta_{n+2,2n}=\binom{n-1}{2}-1$.

Proof.

The assertion follows from [24, Corollary 16] and Theorem 3.10. ∎

Now, we study the position of extremal Betti number of $S/J_{G}$ in terms of the position of extremal Betti number of $S_{H}/J_{H}$.

Proposition 3.12**.**

Let $H$ be a connected graph on the vertex set $[n]$. Let $G=v*H$ be the cone of $v$ on $H$. If $\beta_{i,i+j}^{S_{H}}(S_{H}/J_{H})$ is an extremal Betti number, then $\beta_{i+2,i+2+j}^{S}(S/J_{G})$ is an extremal Betti number and both are equal. Moreover, if $\beta_{k,k+l}^{S}(S/J_{G})$ is an extremal Betti number, then $\beta_{k-2,k+l-2}^{S_{H}}(S_{H}/J_{H})$ is an extremal Betti number and $\beta_{k,k+l}^{S}(S/J_{G})=\beta_{k-2,k+l-2}^{S_{H}}(S_{H}/J_{H})$.

Proof.

Let $\beta_{i,i+j}^{S_{H}}(S_{H}/J_{H})$ be an extremal Betti number of $S_{H}/J_{H}$. Since $H$ is not a complete graph, by Lemma 3.3, $j\geq 2$. If $j\geq 3$, then by Theorem 3.10, $\beta_{i+2,i+2+j}^{S}(S/J_{G})=\beta_{i,i+j}^{S_{H}}(S_{H}/J_{H})$ and for any pair $(r,s)$ with $r\geq i+2$, $s\geq j$ and $(r,s)\neq(i+2,j)$, $\beta_{r,r+s}^{S}(S/J_{G})=0$. Let $p=\operatorname{pd}_{S_{H}}(S_{H}/J_{H})$. If $j=2$, then by Lemma 3.3, $\beta_{p,p+2}^{S_{H}}(S_{H}/J_{H})$ is an extremal Betti number. Therefore it follows from Theorem 3.10 that

[TABLE]

By Remark 3.2, $p\geq n-1$, therefore, $\beta_{p+2,p+4}^{S}(S/J_{G})=\beta_{p,p+2}^{S_{H}}(S_{H}/J_{H})$. Now, let $\beta_{k,k+l}^{S}(S/J_{G})$ is an extremal Betti number. Therefore, by Lemma 3.3, $l\geq 2$. Assume that $l\geq 3$. If possible, $\beta_{k-2,k-2+l}^{S_{H}}(S_{H}/J_{H})$ is not an extremal Betti number. Thus, there exists $r\geq k-2$ and $s\geq l$ such that $(r,s)\neq(k-2,l)$ and $\beta_{r,r+s}^{S_{H}}(S_{H}/J_{H})\neq 0$. Therefore, by virtue of Theorem 3.10, $\beta_{r+2,r+2+s}^{S}(S/J_{G})\neq 0$ which is a contradiction. Hence, $\beta_{k-2,k-2+l}^{S_{H}}(S_{H}/J_{H})$ is an extremal Betti number and by Theorem 3.10, $\beta_{k,k+l}^{S}(S/J_{G})=\beta_{k-2,k-2+l}^{S_{H}}(S_{H}/J_{H})$. Now, if $l=2$, then by Lemma 3.3, $k=\operatorname{pd}_{S}(S/J_{G})$. It follows from Theorem 3.4 that $k=\operatorname{pd}_{S_{H}}(S_{H}/J_{H})+2\geq n+1$. Therefore, by Theorem 3.10,

[TABLE]

Hence, the assertion follows. ∎

Let $H$ be a connected graph on $[n]$. Also, let $G=K_{q}*H$. Then by using Proposition 3.12, we conclude that $S_{H}/J_{H}$ admits unique extremal Betti number if and only if $S/J_{G}$ admits unique extremal Betti number. In particular, if $\beta_{p,p+r}^{S_{H}}(S_{H}/J_{H})$ is an extremal Betti number, then $\beta_{p+2q,p+2q+r}^{S}(S/J_{G})$ is an extremal and $\beta_{p,p+r}^{S_{H}}(S_{H}/J_{H})=\beta_{p+2q,p+2q+r}^{S}(S/J_{G})$.

4. Depth of join of graphs

In this section, we compute the depth of binomial edge ideal of join of two graphs. Let $G_{1}$ and $G_{2}$ be graphs on $[n_{1}]$ and $[n_{2}]$, respectively with $n_{1},n_{2}\geq 2$. We assume that both $G_{1}$ and $G_{2}$ are not complete. The join of $G_{1}$ and $G_{2}$, denoted by $G_{1}*G_{2}$ is the graph with vertex set $[n_{1}]\sqcup[n_{2}]$ and the edge set $E(G_{1}*G_{2})=E(G_{1})\cup E(G_{2})\cup\{\{i,j\}:i\in[n_{1}],j\in[n_{2}]\}$. Let $G=G_{1}*G_{2}$. It follows from [14, Propositions 4.1, 4.5, 4.14] that if $P_{T}(G)$ is a minimal prime of $J_{G}$ for some $T\subseteq[n_{1}]\sqcup[n_{2}]$, then either $T=\emptyset$ or $[n_{1}]\subseteq T$ or $[n_{2}]\subseteq T$. Therefore, by virtue of [5, Theorem 3.2, Corollary 3.9], we have

[TABLE]

Set $Q_{1}=(x_{i},y_{i}:i\in[n_{2}])+J_{G_{1}}$, $Q_{2}=(x_{i},y_{i}:i\in[n_{1}])+J_{G_{2}}$ and $Q_{3}=P_{\emptyset}(G)\cap Q_{2}$. One can see that $Q_{2}+P_{\emptyset}(G)=(x_{i},y_{i}:i\in[n_{1}])+J_{K_{n_{2}}}$ and $Q_{1}+Q_{3}=(x_{i},y_{i}:i\in[n_{2}])+J_{K_{n_{1}}}.$ Thus, we have the following short exact sequences:

[TABLE]

and

[TABLE]

Let $S_{i}=\mathbb{K}[x_{j},y_{j}:j\in[n_{i}]]$ for $i=1,2$. Observe that $0pt_{S}(S/Q_{1})=0pt_{S_{1}}(S_{1}/J_{G_{1}})$, $0pt_{S}(S/(Q_{2}+P_{\emptyset}(G)))=n_{2}+1$, $0pt_{S}(S/(Q_{1}+Q_{3}))=n_{1}+1$ and $0pt_{S}(S/Q_{2})=0pt_{S_{2}}(S_{2}/J_{G_{2}})$. Thus, using Lemma 2.1 in short exact sequence (3),

[TABLE]

and hence from the exact sequence (4) that

[TABLE]

First, we give exact formula for depth of binomial edge ideal of join of two connected graphs.

Theorem 4.1**.**

Let $G=G_{1}*G_{2}$ be join of $G_{1}$ and $G_{2}$, where $G_{1}$ and $G_{2}$ be two connected graphs on vertex sets $[n_{1}]$ and $[n_{2}]$ respectively. Then

[TABLE]

Proof.

First we prove that $0pt_{S}(S/Q_{3})=0pt_{S_{2}}(S_{2}/J_{G_{2}})$. If $0pt_{S_{2}}(S_{2}/J_{G_{2}})<n_{2}+1$, it follows from short exact sequence (3) and Lemma 2.1 that $0pt_{S}(S/Q_{3})=0pt_{S_{2}}(S_{2}/J_{G_{2}})$. Now, we assume that $0pt_{S_{2}}(S_{2}/J_{G_{2}})=n_{2}+1$, by Auslander-Buchsbaum formula, $\operatorname{pd}_{S_{2}}(S_{2}/J_{G_{2}})=n_{2}-1$. Since, $G_{2}$ is not a complete graph, by Lemma 3.3, there exists $j\geq 2$ such that $\beta_{n_{2}-1,n_{2}-1+j}^{S_{2}}(S_{2}/J_{G_{2}})\neq 0$ which implies that $\beta_{p,p+j}^{S}(S/Q_{2})\neq 0$, where $p=2n_{1}+n_{2}-1$. Note that $\operatorname{pd}_{S}(S/P_{\emptyset}(G))=n_{1}+n_{2}-1$ and $\operatorname{pd}_{S}(S/Q_{2})=2n_{1}+n_{2}-1=\operatorname{pd}_{S}(S/((x_{i},y_{i}:i\in[n_{1}])+J_{K_{n_{2}}}))$. Now consider the long exact sequence of Tor in homological degree $p$ corresponding to the short exact sequence (3)

[TABLE]

Since $j\geq 2$, $\beta_{p,p+j}^{S}(S/(P_{\emptyset}(G)+Q_{2}))=0$, which further implies that $\beta_{p,p+j}^{S}(S/Q_{3})\neq 0$. Thus, $\operatorname{pd}_{S}(S/Q_{3})\geq p$ and hence $0pt_{S}(S/Q_{3})\leq 2n_{1}+2n_{2}-p=n_{2}+1$. Hence, by (5), $0pt_{S}(S/Q_{3})=n_{2}+1=0pt_{S_{2}}(S_{2}/J_{G_{2}})$.

Now, if $\min\{0pt_{S_{i}}(S_{i}/J_{G_{i}}):i=1,2\}<n_{1}+1$, then using Lemma 2.1 in short exact sequence (4), we get the desired result. Otherwise, by Remark 3.2, we have $0pt_{S_{1}}(S_{1}/J_{G_{1}})=n_{1}+1=\min\{0pt_{S_{i}}(S_{i}/J_{G_{i}}):i=1,2\}$ and therefore $\operatorname{pd}_{S_{1}}(S_{1}/J_{G_{1}})=n_{1}-1$. Since, $G_{1}$ is not a complete graph, by Lemma 3.3, there exists $l\geq 2$ such that $\beta_{n_{1}-1,n_{1}-1+l}^{S_{1}}(S_{1}/J_{G_{1}})\neq 0$ which further implies that $\beta_{n_{1}+2n_{2}-1,n_{1}+2n_{2}-1+l}^{S}(S/Q_{1})\neq 0$. Note that, $\operatorname{pd}_{S}(S/Q_{1})=n_{1}+2n_{2}-1=\operatorname{pd}_{S}(S/((x_{i},y_{i}:i\in[n_{2}])+J_{K_{n_{1}}}))$. The long exact sequence of Tor in homological degree $q=n_{1}+2n_{2}-1$ and graded degree $q+l$ corresponding to (4) is

[TABLE]

Since, $\beta_{q,q+l}^{S}(S/Q_{1})\neq 0$, we have $\beta_{q,q+l}^{S}(S/J_{G})\neq 0$. Therefore, $\operatorname{pd}_{S}(S/J_{G})\geq q$ and hence $0pt_{S}(S/J_{G})\leq 2n_{1}+2n_{2}-q=n_{1}+1$. It follows from (6) that $0pt_{S}(S/J_{G})\geq n_{1}+1$. Hence, the desired result follows. ∎

We now illustrate our result by the following example. A block of a graph is a maximal nontrivial connected subgraph with no cut vertex. A connected graph is said to be a block graph if every block of that graph is a complete graph.

Example 4.2**.**

If $G_{1}$ be a connected block graph and $G_{2}=C_{n_{2}}$ with $n_{1}\geq n_{2}\geq 4$, then by virtue of [3, Theorem 1.1] $0pt_{S_{1}}(S_{1}/J_{G_{1}})=n_{1}+1$. By [24, Corollary 16] that $0pt_{S_{2}}(S_{2}/J_{G_{2}})=n_{2}$. Hence, $0pt_{S}(S/J_{G_{1}*G_{2}})=0pt_{S_{2}}(S_{2}/J_{G_{2}})=n_{2}$.

Now, we move on to study the join of a connected graph and a disconnected graph.

Theorem 4.3**.**

Let $G_{1}$ be a connected graph on the vertex set $[n_{1}]$ and $G_{2}$ be a disconnected graph on the vertex set $[n_{2}]$. Then

[TABLE]

Proof.

We claim that $0pt_{S}(S/Q_{3})=\min\{0pt_{S_{2}}(S_{2}/J_{G_{2}}),n_{2}+2\}$. First assume that $n_{2}+1<0pt_{S_{2}}(S_{2}/J_{G_{2}})$. Therefore, the claim follows from the short exact sequence (3) and Lemma 2.1. If $n_{2}+1>0pt_{S_{2}}(S_{2}/J_{G_{2}})$, it follows from short exact sequence (3) that $0pt_{S}(S/Q_{3})=0pt_{S_{2}}(S_{2}/J_{G_{2}})$. Now, we assume that $0pt_{S_{2}}(S_{2}/J_{G_{2}})=n_{2}+1$ which implies that $\operatorname{pd}_{S_{2}}(S_{2}/J_{G_{2}})=n_{2}-1$. Since, $G_{2}$ is a disconnected graph, by Remark 3.8, there exists $j\geq 2$ such that $\beta_{n_{2}-1,n_{2}-1+j}^{S_{2}}(S_{2}/J_{G_{2}})\neq 0$ which further implies that $\beta_{p,p+j}^{S}(S/Q_{2})\neq 0$, where $p=2n_{1}+n_{2}-1$. Note that $\operatorname{pd}_{S}(S/P_{\emptyset}(G))=n_{1}+n_{2}-1$, $\operatorname{pd}_{S}(S/Q_{2})=2n_{1}+n_{2}-1=\operatorname{pd}_{S}(S/((x_{i},y_{i}:i\in[n_{1}])+J_{K_{n_{2}}}))$. Now, consider the long exact sequence of Tor (7). Since $\beta_{p,p+j}^{S}(S/Q_{2})\neq 0$, we have that $\beta^{S}_{p,p+j}(S/Q_{3})\neq 0$. Thus, $\operatorname{pd}_{S}(S/Q_{3})\geq p$ and hence $0pt_{S}(S/Q_{3})\leq 2n_{1}+2n_{2}-p=n_{2}+1$. Therefore, by (5), $0pt_{S}(S/Q_{3})=n_{2}+1=0pt_{S_{2}}(S_{2}/J_{G_{2}})$. Hence, we have

[TABLE]

Now, if $\min\{0pt_{S_{1}}(S_{1}/J_{G_{1}}),0pt_{S_{2}}(S_{2}/J_{G_{2}}),n_{2}+2\}<n_{1}+1$, then by applying Lemma 2.1 in short exact sequence (4), we get the desired result. Otherwise, we have $\min\{0pt_{S_{1}}(S_{1}/J_{G_{1}}),0pt_{S_{2}}(S_{2}/J_{G_{2}}),n_{2}+2\}\geq n_{1}+1$ and hence by Remark 3.2, $0pt_{S_{1}}(S_{1}/J_{G_{1}})=n_{1}+1$. Therefore $\operatorname{pd}_{S_{1}}(S_{1}/J_{G_{1}})=n_{1}-1$. Since, $G_{1}$ is not a complete graph, by Lemma 3.3, there exists $l\geq 2$ such that $\beta_{n_{1}-1,n_{1}-1+l}^{S_{1}}(S_{1}/J_{G_{1}})\neq 0$ which implies that $\beta_{q,q+l}^{S}(S/Q_{1})\neq 0$, where $q=n_{1}+2n_{2}-1$. Note that $\operatorname{pd}_{S}(S/Q_{1})=q=\operatorname{pd}_{S}(S/((x_{i},y_{i}:i\in[n_{2}])+J_{K_{n_{1}}}))$. Consider, the long exact sequence of Tor (8). Since, $\beta_{q,q+l}^{S}(S/Q_{1})\neq 0$, we have $\beta_{q,q+l}^{S}(S/J_{G})\neq 0$. Therefore, $\operatorname{pd}_{S}(S/J_{G})\geq q$ and hence $0pt_{S}(S/J_{G})\leq 2n_{1}+2n_{2}-q=n_{1}+1$. It follows from (6) that $0pt_{S}(S/J_{G})\geq n_{1}+1$. Hence, the assertion follows. ∎

We now compute the depth of the binomial edge ideal of join of two disconnected graphs.

Theorem 4.4**.**

Let $G=G_{1}*G_{2}$ be the join of $G_{1}$ and $G_{2}$, where $G_{1}$ and $G_{2}$ are disconnected graphs on $[n_{1}]$ and $[n_{2}]$ respectively. Assume that $n_{2}\geq n_{1}$. Then

[TABLE]

Proof.

It follows from the proof of Theorem 4.3 that

[TABLE]

Now, if $\min\{0pt_{S_{1}}(S_{1}/J_{G_{1}}),0pt_{S_{2}}(S_{2}/J_{G_{2}}),n_{2}+2\}<n_{1}+1$, then using Lemma 2.1 in short exact sequence (4), we get the desired result.

If $\min\{0pt_{S_{1}}(S_{1}/J_{G_{1}}),0pt_{S_{2}}(S_{2}/J_{G_{2}}),n_{2}+2\}=n_{1}+1$, then either $0pt_{S_{1}}(S_{1}/J_{G_{1}})=n_{1}+1$ or $0pt_{S}(S/Q_{3})=n_{1}+1$. Now, if $0pt_{S_{1}}(S_{1}/J_{G_{1}})=n_{1}+1$, then by Auslander-Buchsbaum formula, $\operatorname{pd}_{S_{1}}(S_{1}/J_{G_{1}})=n_{1}-1$. Therefore, by virtue of Remark 3.8 there exists $j\geq 2$ such that $\beta_{n_{1}-1,n_{1}-1+j}^{S_{1}}(S_{1}/J_{G_{1}})\neq 0$ which implies that $\beta_{q,q+j}^{S}(S/Q_{1})\neq 0$, where $q=n_{1}+2n_{2}-1$. Note that $\operatorname{pd}_{S}(S/Q_{1})=q=\operatorname{pd}_{S}(S/((x_{i},y_{i}:i\in[n_{2}])+J_{K_{n_{1}}}))$. Consider, the long exact sequence of Tor (8) in graded degree $q+j$. Since, $\beta_{q,q+j}^{S}(S/Q_{1})\neq 0$, we have $\beta_{q,q+j}^{S}(S/J_{G})\neq 0$. Therefore, $\operatorname{pd}_{S}(S/J_{G})\geq q$ and hence $0pt_{S}(S/J_{G})\leq 2n_{1}+2n_{2}-q=n_{1}+1$. Now, the assertion follows from (6). Assume now that $0pt_{S}(S/Q_{3})=n_{1}+1$. Since, $n_{1}\leq n_{2}$, $0pt_{S}(S/Q_{3})=n_{1}+1=0pt_{S_{2}}(S_{2}/J_{G_{2}})=0pt_{S}(S/Q_{2})$. Note that $\operatorname{pd}_{S}(S/Q_{3})=q=\operatorname{pd}_{S}(S/Q_{2})$. Since, $G_{2}$ is a disconnected graph and $0pt_{S_{2}}(S_{2}/J_{G_{2}})=n_{1}+1\leq n_{2}+1$, either $G_{2}$ has atleast two nontrivial components or $G_{2}$ has one nontrivial component which is not complete. In first case, by Lemma 3.7, there exists $j\geq 2$ such that $\beta_{2n_{2}-n_{1}-1,2n_{2}-n_{1}-1+j}^{S_{2}}(S_{2}/J_{G_{2}})\neq 0$ which further implies that $\beta_{q,q+j}^{S}(S/Q_{2})\neq 0$. If $G_{2}$ has exactly one nontrivial component say $H$, then $\operatorname{pd}_{S_{2}}(S_{2}/J_{G_{2}})=\operatorname{pd}_{S_{H}}(S_{H}/J_{H})=2n_{2}-n_{1}-1$, where $S_{H}=\mathbb{K}[x_{j},y_{j}:j\in V(H)]$. Now, by Lemma 3.3, there exists $j\geq 2$ such that $\beta_{2n_{2}-n_{1}-1,2n_{2}-n_{1}-1+j}^{S_{2}}(S_{2}/J_{G_{2}})\neq 0$ which further implies that $\beta_{q,q+j}^{S}(S/Q_{2})\neq 0$. The long exact sequence of Tor corresponding to (3) in homological degree $q$ and graded degree $q+j$ is

[TABLE]

Therefore, $\beta_{q,q+j}^{S}(S/Q_{3})\neq 0$. Thus, it follows from (8) that $\beta_{q,q+j}^{S}(S/J_{G})\neq 0$. Therefore, $\operatorname{pd}_{S}(S/J_{G})\geq q$ and hence, $0pt_{S}(S/J_{G})\leq 2n_{1}+2n_{2}-q=n_{1}+1$. Now, along with (6), we get the assertion.

Also, if $\min\{0pt_{S_{1}}(S_{1}/J_{G_{1}}),0pt_{S_{2}}(S_{2}/J_{G_{2}}),n_{2}+2\}>n_{1}+1$, then again using Lemma 2.1 in the short exact sequence (4), $0pt_{S}(S/J_{G})=n_{1}+2$. Hence, the desired result follows. ∎

As an immediate consequence, we obtain the depth of complete multipartite graph.

Corollary 4.5**.**

Let $G=K_{n_{1},\cdots,n_{k}}$ be a complete multipartite graph with $2\leq n_{1}\leq\cdots\leq n_{k}$. Then $0pt_{S}(S/J_{G})=n_{1}+2.$

5. Construction of Graph

In this section, we construct a graph $G$ such that $\operatorname{reg}(S/J_{G})=r$ and the number of extremal Betti numbers of $S/J_{G}$ is $b$, where $1\leq b<r$. We now set some notation for the rest of this section. Let $G_{1}$ and $G_{2}$ be two connected graphs which are not complete on the vertex sets $[n_{1}]$ and $[n_{2}]$, respectively. Let $p_{i}=\operatorname{pd}_{S_{i}}(S_{i}/J_{G_{i}})$ and $r_{i}=\operatorname{reg}(S_{i}/J_{G_{i}})$ for $i=1,2$. By Remark 3.2, $p_{i}\geq n_{i}-1$, for $i=1,2$.

Lemma 5.1**.**

Let $G_{1}$ and $G_{2}$ be graphs on $[n_{1}]$ and $[n_{2}]$, respectively. Let $G=G_{1}*G_{2}$. If $\operatorname{reg}(S/J_{G})=2$, then $S/J_{G}$ admits unique extremal Betti number.

Proof.

Proof follows from the Lemma 3.3. ∎

We consider the long exact sequence of Tor corresponding to the exact sequence (3)

[TABLE]

where $Q_{2}=(x_{i},y_{i}:i\in[n_{1}])+J_{G_{2}}$ and $Q_{3}=P_{\emptyset}(G)\cap Q_{2}$.

Now, we prove that extremal Betti numbers of $S/Q_{3}$ and $S/Q_{2}$ coincide in terms of position and value.

Lemma 5.2**.**

Let $G=G_{1}*G_{2}$ be the join graph on $[n_{1}]\sqcup[n_{2}]$. If $\beta_{k,k+l}^{S_{2}}(\frac{S_{2}}{J_{G_{2}}})$ is an extremal Betti number, then $\beta_{k+2n_{1},k+2n_{1}+l}^{S}(\frac{S}{Q_{3}})$ is an extremal Betti number. Moreover, extremal Betti number of $S/Q_{3}$ is of the form $\beta_{k+2n_{1},k+2n_{1}+l}^{S}(\frac{S}{Q_{3}})$, where $\beta_{k,k+l}^{S_{2}}(\frac{S_{2}}{J_{G_{2}}})$ is an extremal Betti number.

Proof.

It follows from proof of Theorem 4.1 that $0pt_{S}(S/Q_{3})=0pt_{S_{2}}(S_{2}/J_{G_{2}})$. Thus, $\operatorname{pd}_{S}(S/Q_{3})=p_{2}+2n_{1}$. Since, $\operatorname{pd}_{S}(S/P_{\emptyset}(G))=n_{1}+n_{2}-1$ and $\operatorname{pd}_{S}(S/((x_{i},y_{i}:i\in[n_{1}])+J_{K_{n_{2}}}))=2n_{1}+n_{2}-1$, by Lemma 3.3,

[TABLE]

Also, $\operatorname{Tor}_{p_{2}+2n_{1}+1,p_{2}+2n_{1}+1}^{S}(S/((x_{i},y_{i}:i\in[n_{1}])+J_{K_{n_{2}}}))=0$. By considering the long exact sequence of Tor (9) in homological degree $p_{2}+2n_{1}$ and graded degree $p_{2}+2n_{1}+1$, we get, $\beta_{p_{2}+2n_{1},p_{2}+2n_{1}+1}^{S}(\frac{S}{Q_{3}})=0$. Now, let $\beta_{k,k+l}^{S_{2}}(\frac{S_{2}}{J_{G_{2}}})$ is an extremal Betti number. By virtue of Lemma 3.3, $l\geq 2$ and therefore the long exact sequence of Tor (9) in homological degree $p=k+2n_{1}$ and graded degree $p+l$ is

[TABLE]

Now if $l>2$, then we have

[TABLE]

Thus, $\beta_{p,p+l}^{S}(\frac{S}{Q_{3}})\neq 0$. Let $(s,t)\neq(p,l)$ with $s\geq p,t\geq l$. Taking homological degree $s\geq p$ and graded degree $s+t\geq p+l$ in (9), we have

[TABLE]

Note that $s-2n_{1}\geq k,t\geq l$ and $(s-2n_{1},t)\neq(k,l)$. Therefore,

[TABLE]

Hence, $\beta_{p,p+l}^{S}(\frac{S}{Q_{3}})$ is an extremal Betti number. Now we assume that $l=2$. It follows from Lemma 3.3 that $k=p_{2}$. Note that $\operatorname{pd}_{S}(S/((x_{i},y_{i}:i\in[n_{1}])+J_{K_{n_{2}}}))=2n_{1}+n_{2}-1\leq p_{2}+2n_{1}$. Now, consider the long exact sequence of Tor (9) in homological degree $p=p_{2}+2n_{1}$, $\operatorname{Tor}_{p_{2}+2n_{1},p_{2}+2n_{1}+j}^{S}\left(\frac{S}{Q_{3}},\mathbb{K}\right)\simeq\operatorname{Tor}_{p_{2},p_{2}+j}^{S_{2}}\left(\frac{S_{2}}{J_{G_{2}}},\mathbb{K}\right),$ for $j\geq 2$. Since, $\beta_{p_{2},p_{2}+2}^{S_{2}}(\frac{S_{2}}{J_{G_{2}}})$ is an extremal Betti number, $\beta_{p,p+2}^{S}(\frac{S}{Q_{3}})$ is an extremal Betti number.

Now, let $\beta_{i,i+j}^{S}(\frac{S}{Q_{3}})$ be an extremal Betti number. Consider, the long exact sequence of Tor (9) in homological degree $i$ and graded degree $i+j$. If $j>2$, then

[TABLE]

Therefore, $\beta_{i-2n_{1},i-2n_{1}+j}^{S_{2}}(\frac{S_{2}}{J_{G_{2}}})\neq 0$. Now, if for some $s\geq i-2n_{1},t\geq j$ with $(s,t)\neq(i-2n_{1},j)$, $\beta_{s,s+t}^{S_{2}}(\frac{S_{2}}{J_{G_{2}}})\neq 0$. Then, $\beta_{s+2n_{1},s+2n_{1}+t}^{S}(\frac{S}{Q_{3}})\neq 0$ and $s+2n_{1}\geq i,t\geq j$ and $(s+2n_{1},t)\neq(i,j)$ which is a contradiction. Hence, $\beta_{i-2n_{1},i-2n_{1}+j}^{S_{2}}(\frac{S_{2}}{J_{G_{2}}})$ is an extremal Betti number. If $j=2$, then by Lemma 3.3, $i=p_{2}+2n_{1}$. For any $l\geq 2$, we have $\operatorname{Tor}_{p_{2}+2n_{1},p_{2}+2n_{1}+l}^{S}\left(\frac{S}{Q_{3}},\mathbb{K}\right)\simeq\operatorname{Tor}_{p_{2},p_{2}+l}^{S_{2}}\left(\frac{S_{2}}{J_{G_{2}}},\mathbb{K}\right)$. Therefore, $\beta_{p_{2},p_{2}+2}^{S_{2}}(\frac{S_{2}}{J_{G_{2}}})$ is an extremal Betti number. This completes the proof. ∎

It follows from above theorem that $\operatorname{reg}(S/Q_{3})=\operatorname{reg}(S/Q_{2})=\operatorname{reg}(S_{2}/J_{G_{2}})$. Assume that $2\leq r_{1}\leq r_{2}$ and $p_{1}\leq p_{2}$. Since, $G_{1}$ is a connected graph with $r_{1}\geq 2$, by Lemma 3.3, if $\beta_{k,k+l}^{S_{1}}(\frac{S_{1}}{J_{G_{1}}})$ is an extremal Betti number, then $l\geq 2$. We now consider long exact sequence of Tor in homological degree $k$ and graded degree $k+l\geq k+2$ corresponding to the exact sequence (4),

[TABLE]

where $Q_{1}=(x_{i},y_{i}:i\in[n_{2}])+J_{G_{1}}$.

Lemma 5.3**.**

Let $G=G_{1}*G_{2}$ be the join graph on $[n_{1}]\sqcup[n_{2}]$. Suppose $0pt_{S_{1}}(S_{1}/J_{G_{1}})<0pt_{S_{2}}(S_{2}/J_{G_{2}})$ i.e., $p_{2}+2n_{1}<p_{1}+2n_{2}$. If $\beta_{k,k+l}^{S}(S/J_{G})$ is an extremal Betti number, then

[TABLE]

Proof.

Note that $\operatorname{pd}_{S}(S/((x_{i},y_{i}:i\in[n_{2}])+J_{K_{n_{1}}}))=2n_{2}+n_{1}-1<2n_{2}+p_{1}+1,$ by Remark 3.2. Therefore, $\operatorname{Tor}_{p_{1}+2n_{2}+1,p_{1}+2n_{2}+1+j}^{S}\left(\frac{S}{(x_{i},y_{i}:i\in[n_{2}])+J_{K_{n_{1}}}},\mathbb{K}\right)=0$, for $j\geq 1$. If $l=2$, by Lemma 3.3, $k=\operatorname{pd}_{S}(S/J_{G})=p_{1}+2n_{2}$. Also if $l>2$, then

[TABLE]

Hence, by (10)

[TABLE]

∎

By [21, Theorem 2.1], $\operatorname{reg}(S/J_{K_{n}})=1$ and hence $S/J_{K_{n}}$ admits unique extremal Betti number. So for $r=b=1$, consider $G=K_{n}$. Now assume that $r\geq 2$. It follows from Betti diagram and Lemmas 3.3, 3.7 that $b\leq r-1$.

Theorem 5.4**.**

Let $r$ and $b$ be two positive integers with $1\leq b\leq r-1$. Then there exists a graph $G=G_{r,b}$ such that $\operatorname{reg}(S/J_{G})=r$ and the number of extremal Betti numbers of $S/J_{G}$ is $b$.

Proof.

Take $G=G_{r,b}=P_{r-b+2}*\cdots*P_{r+1}$. Note that $n=|V(G)|=br-\frac{b(b-3)}{2}$ and by recursively applying Theorem 4.1, $0pt_{S}(S/J_{G})=r-b+3$. Now, by Auslander-Buchsbaum formula, $p=\operatorname{pd}_{S}(S/J_{G})=(2b-1)r-(b-1)(b-3)$. It follows from [22, Theorem 2.1] that $\operatorname{reg}(S/J_{G})=r.$ We now prove that extremal Betti numbers of $S/J_{G}$ are precisely $\beta^{S}_{p-i,p+r-b+1}\left(\frac{S}{J_{G}}\right)=1$, for $0\leq i\leq b-1$. We proceed by induction on $b$. If $b=1$, then $G=G_{r,1}=P_{r+1}$. By [3, Corollary 1.2], $J_{G}$ is complete intersection ideal, $S/J_{G}$ has a unique extremal Betti number, $\beta_{r,2r}^{S}\left(\frac{S}{J_{G}}\right)=1$. Now, assume that $b>1$ and extremal Betti numbers of $S_{2}/J_{G_{2}}$ are precisely $\beta^{S_{2}}_{p_{2}-i,p_{2}+r-b+2}\left(\frac{S_{2}}{J_{G_{2}}}\right)$ for $0\leq i\leq b-2$, where $G_{2}=G_{r,b-1}=P_{r-b+3}*\cdots*P_{r+1}$, $S_{2}=\mathbb{K}[x_{j},y_{j}:j\in V(G_{2})]$ and $p_{2}=\operatorname{pd}_{S_{2}}(S_{2}/J_{G_{2}})$. Observe that $n_{2}=|V(G_{2})|=(b-1)r-\frac{(b-1)(b-4)}{2}$. Also, by Theorem 4.1, $0pt_{S_{2}}(S_{2}/J_{G_{2}})=r-b+4$. Thus, by Auslander-Buchsbaum formula, $p_{2}=2n_{2}-r+b-4=(2b-3)r-(b-2)(b-4).$ Set $G_{1}=P_{r-b+2}$, $n_{1}=|V(G_{1})|=r-b+2$ and $S_{1}=\mathbb{K}[x_{j},y_{j}:j\in V(G_{1})]$. By [3, Corollary 1.2], $p_{1}=\operatorname{pd}_{S_{1}}({S_{1}}/{J_{G_{1}}})=r-b+1=r_{1}=\operatorname{reg}(S_{1}/J_{G_{1}})$. Note that $G=G_{r,b}=G_{1}*G_{2}$. Since $J_{G_{1}}$ is complete intersection ideal, $\beta^{S_{1}}_{p_{1},p_{1}+r-b+1}\left(\frac{S_{1}}{J_{G_{1}}}\right)=1$ is the extremal Betti number of $S_{1}/J_{G_{1}}$. Therefore, $\beta^{S}_{p_{1}+2n_{2},p_{1}+2n_{2}+r-b+1}\left(\frac{S}{Q_{1}}\right)$ is unique extremal Betti number of $S/Q_{1}$. Note that $p_{1}+2n_{2}=(2b-1)r-(b-1)(b-3)=p$.

Let $\beta_{k,k+l}^{S}(S/J_{G})$ is an extremal Betti number. Now, by Lemma 5.3, we have

[TABLE]

It follows from Lemma 5.2 that extremal Betti numbers of $S/Q_{3}$ are $\beta^{S}_{p_{2}+2n_{1}-i,p_{2}+2n_{1}+r-b+2}\left(\frac{S}{Q_{3}}\right)$, for $0\leq i\leq b-2$. Note that $p_{2}+2n_{1}=p_{2}+2(r-b+2)=p-1$. Therefore, extremal Betti numbers of $S/Q_{3}$ are $\beta^{S}_{p-i-1,p+r-b+1}\left(\frac{S}{Q_{3}}\right)=1$, for $0\leq i\leq b-2$. Since $b\leq r-1$, $r-b+1\geq 2$. So, for $j\geq r-b+2\geq 3$ and $1\leq k\leq p$,

[TABLE]

Therefore, $\beta_{p-i-1,p+r-b+1}^{S}\left(\frac{S}{J_{G}}\right)=1$, for $0\leq i\leq b-2$ are extremal Betti numbers of $S/J_{G}$. Now, it remains to prove that $\beta_{p,p+r-b+1}^{S}\left(\frac{S}{J_{G}}\right)$ is an extremal Betti number.

Taking long exact sequence of Tor (9) in homological degree $p$ and graded degree $p+j=p+r-b+1\geq p+2$, we have $\operatorname{Tor}^{S}_{p,p+r-b+1}\left(\frac{S}{J_{G}}\right)\simeq\operatorname{Tor}^{S}_{p,p+r-b+1}\left(\frac{S}{Q_{1}}\right)\simeq\operatorname{Tor}^{S_{1}}_{p_{1},p_{1}+r-b+1}\left(\frac{S_{1}}{J_{G_{1}}}\right)$. Therefore, $\beta^{S}_{p,p+r-b+1}\left(\frac{S}{J_{G}}\right)=1$ is also an extremal Betti number. Hence the number of extremal Betti numbers of $S/J_{G}$ is $b$ and the extremal Betti number are of the form $\beta^{S}_{p-i,p+r-b+1}\left(\frac{S}{J_{G}}\right)=1$, for $0\leq i\leq b-1$. ∎

Observe that the projective dimension of $S/J_{G_{r,b}}$ is very large. Hence the following question arises:

Question 5.5**.**

Does there exist a graph $G$ such that the projective dimension is bounded by a linear function of $b$ and $r$, where $r=\operatorname{reg}(S/J_{G})$ and $b$ is the number of extremal Betti numbers of $S/J_{G}$?

Acknowledgement: The authors are grateful to their advisor A. V. Jayanthan for constant support, valuable ideas and suggestions. The first author thanks the National Board for Higher Mathematics, India for the financial support. The second author thanks University Grants Commission, Government of India for the financial support.