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\title{\bf On the Betti numbers of some classes\\ of binomial edge ideals}

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\author{Sohail Zafar\\
\small Mathematics Department\\[-0.8ex]
\small University of Management and Technology\\[-0.8ex]
\small Lahore, Pakistan\\
\small\tt sohailahmad04@gmail.com \\
\and
Zohaib Zahid\thanks{Research supported by HEC Pakistan.}\\
\small Abdus Salam School of Mathematical Sciences\\[-0.8ex]
\small Government College University\\[-0.8ex]
\small Lahore, Pakistan\\
\small\tt zohaib\_zahid@hotmail.com}

% \date{\dateline{submission date}{acceptance date}\\
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\date{\dateline{Sep 3, 2013}{Dec 13, 2013}{Dec 20, 2013}\\
\small Mathematics Subject Classifications: 05E40, 16E30}

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
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\begin{abstract}
We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we
give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs
depending on induced subgraphs.

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} Castelnuovo-Mumford regularity, Betti numbers, binomial edge ideal.
\end{abstract}

\section{Introduction}
Let $K$ denote a field. Let $G$ denote a connected, simple and undirected graph over the
vertices labeled by $[n]=\{1,2,\dots,n\}.$ The binomial edge ideal $J_{G}\subseteq
S=K[x_{1},\dots,x_{n},y_{1},\dots\\,y_{n}]$ is an ideal generated by all binomials $
x_{i}y_{j}-x_{j}y_{i}$ , $i<j$ , such that $\{i,j\}$ is an edge of $G$. It was introduced in \cite{HKR} and independently at the same time in \cite{MO}. It is a natural generalization of the notion of monomial edge ideal which is introduced by Villarreal in \cite{Vi}.


The main purpose of this paper is to study the minimal free resolution of certain classes of binomial edge ideals. The arithmetic properties of binomial edge ideals in terms of combinatorial properties of graphs (and vice versa) were studied by many authors in \cite{HKR}, \cite{MO}, \cite{EHH}, \cite{So}, \cite{ps}, \cite{AR}, \cite{GR}, \cite{MM}, and \cite{SD}. The reduced Gr\H{o}bner basis and minimal primary decomposition of binomial edge ideal was given in the paper of Herzog et al. \cite{HKR}. The Cohen-Macaulay property of binomial edge ideal were studied in \cite{EHH}, \cite{AR} and \cite{GR}. As a certain generalization of the Cohen-Macaulay property the second author has studied approximately Cohen-Macaulay property in \cite{So}.


There is not so much work done so far in the direction of the Betti numbers and Castelnuovo-Mumford regularity of binomial edge ideals. The minimal free resolution of the binomial edge ideal of simplest classes (complete graph and line graph) is well known. In \cite{SD}, the authors
determine the initial Betti number of the binomial edge ideal of an arbitrary graph. In fact they shows that $\beta_{2,3}=2l$ where $l$ is the total number of 3-cycles in the graph $G$. They also discussed the vanishing and non-vanishing of certain Betti numbers. In \cite{ps}, there is a computation of the Castelnuovo-Mumford regularity and all the Betti numbers in the case of complete bipartite graph. The relationship between the Betti numbers of a graph and the Betti numbers of its induced subgraph (see Theorem \ref{p4}) was recently shown in \cite{MM} by Matsuda and Murai. They also give the Castelnuovo-Mumford regularity bounds for binomial edge ideals, namely $\ell -1 \leq \reg (S/J_G) \leq n-1,$ where $\ell$ denotes the number of vertices of the longest induced line graph of $G$.


In the present paper we compute the Castelnuovo-Mumford regularity and all the Betti numbers in the case of cycle graphs and two more classes of graphs which we denote by $\mathcal{T}_3$ and $\mathcal{G}_3$ (see Definitions \ref{d1} and \ref{d2}). In all of our classes Castelnuovo-Mumford regularity is quite large.  As an application of our investigation we improve the lower bounds for the Castelnuovo-Mumford regularity of an arbitrary graph by applying as it was done in \cite{MM}.


The paper is organized as follows:
In Section 2, we introduce some notation and give some results that we need in the rest of the paper. In particular we give a short summary on minimal free resolutions. In Section 3, we compute the Castelnuovo-Mumford regularity and the Betti numbers of the binomial edge ideal associated with a cycle graph and obtain a lower bound for the Castelnuovo-Mumford regularity of an arbitrary graph. In Section 4, we do the same for the classes of graphs $\mathcal{T}_3$ and $\mathcal{G}_3$ as we did for the cycle in Section 3.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
In this section we will introduce the notation used in the article. Moreover we
summarize a few auxiliary results that we need.

We denote by $G$ a connected undirected graph on $n$ vertices labeled by
$[n] = \{1,2,\ldots,\\n\}$. For an arbitrary field $K$ let $S = K[x_1,\dots,x_n,y_1,\dots,y_n]$
denote the polynomial ring in the $2n$ variables. To the
graph $G$ one can associate an ideal $J_G \subset S$ generated by all binomials
$x_iy_j-x_jy_i$ for $i<j$ such that $\{i,j\}$ forms an edge of $G$. This Ideal $J_G$ is called \textbf{binomial edge ideal} associated to the graph $G$.
This construction was invented by Herzog et al. in \cite{HKR} and independently found in \cite{MO}. At first let us
recall some of their definitions.
\begin{definition}
For a subset $W\subset[n]$, a graph $G_W$ on vertex set $W$ is called induced subgraph of $G$ if for all $i,j\in W$, $\{i,j\}$ is an edge of $G_W$ if and only if $\{i,j\}$ is an edge of $G$.
\end{definition}
\begin{definition} \label{p1} Fix the previous notation. For a set $T \subset [n]$,
let $G_{[n]\setminus{T}}$ is the induced subgraph of $G$ with vertex set $[n]\setminus{T}$.
Let $c = c(T)$ denote the number of connected components of $G_{[n]\setminus{T}}$.
Let $G_1,\ldots,G_c$ denote the connected components of $G_{[n]\setminus{T}}$. Then define
\[
P_{T}(G)=(\cup _{i\in T}\{x_{i},y_{i}\},J_{\tilde{G}_{1}},\dots,J_{\tilde{G}%
_{C(T)}}),
\]
where $\tilde{G}_i, i =1,\ldots,c,$ denotes the complete graph on the vertex set
of the connected component $G_i, i = 1,\ldots,c$.
\end{definition}

The following result whose proof can be found in Section 3 of the paper \cite{HKR} is important for the understanding of the binomial edge ideal of $G$.
\begin{lemma} \label{p2}
With the previous notation the following holds:
\begin{itemize}
\item[(a)] $P_{T}(G)\subset S$
is a prime ideal of height $n-c+|T|,$ where $|T|$ denotes the number of
elements of $T$.
\item[(b)] $J_{G}=\cap _{T \subseteq [n]}P_{T}(G).$
\item[(c)] $J_{G}\subset P_{T}(G)$ is a
minimal prime if and only if either  $T=\emptyset $ \ or $T\neq \emptyset $
and $c(T \setminus \{i\})<c(T)$  for each $i \in T$.
\end{itemize}
\end{lemma}


Therefore $J_{G}$ is the intersection of prime ideals. That is, $S/J_{G}$ is
a reduced ring. Moreover, we remark that $J_G$ is an ideal generated by quadrics
and therefore homogeneous, so that $S/J_{G}$ is a graded ring with natural grading induced
by the $\mathbb{N}$-grading of $S$.
\begin{remark} If we define a grading on $S$ by setting $\deg x_i=\deg y_i=e_i$ where $e_i$ is the $i-$th unit vector of $\mathbb{N}^n$ then $S/J_{G}$ is $\mathbb{N}^n$-graded too.
\end{remark}
Let $M$ a graded finitely generated $S$-module. By Hilbert's Syzygy Theorem, $M$ has a finite minimal graded free resolution:
\[
F_{\bullet}: 0\to F_p\to \cdots \to F_1\to F_0\to M \to 0
\]
where $F_i=\bigoplus_jS(-d_{i,j})^{\beta_{i,j}}$ for $i\geq 0$ and $p$ is called the \textbf{projective dimension} of $M$.
The numbers $\beta_{ij}$ are uniquely determined by $M$ i.e. $\beta_{i,j}(M) = \dim_K \Tor_i^S(K,M)_{j}, i,j \in \mathbb{Z},$ as \textbf{graded Betti numbers} of $M$. We can also define \textbf{Castelnuovo-Mumford regularity} $\reg M = \max \{j-i \in \mathbb{Z} | \beta_{i,j}(M) \not= 0\}$. The \textbf{Betti table} looks as in the following:
\[
\begin{array}{c|ccccc}
   & 0 & 1  & \cdots & p \\
\hline
0 & \beta_{0,0} & \beta_{1,1} & \cdots & \beta_{p,p} \\
1 & \beta_{0,1} & \beta_{1,2} & \cdots & \beta_{p,p+1} \\
\vdots & \vdots & \vdots & \vdots \\
r & \beta_{0,r} & \beta_{1,1+r} & \cdots & \beta_{p,p+r}
\end{array}
\] Note that all the $\beta_{i,j}$ outside
of the Betti table are zero. For more details and related facts we refer the book of Burns and Herzog \cite{BH}. The following result in \cite{BH} is important for us.
\begin{lemma}\label{xyz} Let $M$ denote a finitely generated graded $S$-module then Hilbert series of $M$ can be computed from the graded Betti numbers as follows:
\[ H(M,t)=\frac{\sum_{-\infty< j < \infty} \sum_{i=0}^{2n}(-1)^i \beta_{i,j}(M)t^j}{(1-t)^{2n}}.\]
\end{lemma}
\begin{definition} \label{p3} Let $M$ denote a finitely generated graded
$S$-module and $d = \dim M$. For an integer $i \in \mathbb{Z}$ put
\[
\omega^i(M) = \Ext_S^{2n-i}(M,S(-2n))
\]
and call it the \textbf{$i$-th module of deficiency}. The module $\omega(M):= \omega^d(M)$ is called
the \textbf{canonical module} of $M$.
\end{definition}
These modules have been introduced and studied
in \cite{Sch1}.
The following theorem whose proof can be found in \cite[Corollary 3.3.9]{BH} is important for us.
\begin{theorem}\label{p5} Let $M$ denote a finitely generated graded Cohen-Macaulay $S$-module of projective dimension $p = 2n - \dim M$. Let
\[
F_{\bullet}: 0\to F_p\to \cdots \to F_1\to F_0\to 0
\] be the minimal free resolution of $M$. Let $G_{\bullet}=\Hom_S(F_{\bullet},S(-2n))$ be the dual complex
\[
G_{\bullet}: 0\to G_p\to \cdots \to G_1\to G_0\to 0
\] where $G_i=\Hom(F_{p-i},S(-2n))$ for i=0,\ldots,p. Then $G_{\bullet}$ is the minimal free resolution of $\omega(M).$
\end{theorem}
 Recently K. Matsuda and S. Murai in \cite[Corollary 2.2]{MM} proved the relationship between the Betti numbers of the graph with the Betti numbers of its induced subgraph. The result is as follows:
\begin{theorem}\label{p4}
Let $G_W$ be an induced subgraph of $G$. Then $\beta_{i,j}(S/J_G)\geq \beta_{i,j}(S/J_{G_W})$ for all $i,j$.
\end{theorem}

\section{Betti Numbers of the binomial edge ideal of a cycle}
\begin{definition}
A \textbf{cycle} is a graph in which all the vertices are of degree 2.
\end{definition}
In particular, for $n=3$ it is triangle and $n=4$ it is square.
We denote the cycle on vertex set $[n]$ by $C$ and its binomial edge ideal by $J_C$. It is known from \cite[Theorem 4.5]{So} that, $S/J_C$ is approximately Cohen-Macaulay ring of $\dim(S/J_C)=n+1$.

\begin{theorem}\label{c1}Let $J_L$ be the binomial edge ideal of a
line $L$ on the vertex set $[n]$ , $g=x_{1}y_{n}-x_{n}y_{1}$ and $J_{\tilde{G}}$
be binomial edge ideal of the complete graph on $[n]$ then:
\begin{enumerate}
  \item [(a)] $J_L:g/J_L\cong \omega (S/J_{\tilde{G}})(2)$
  \item [(b)] The Hilbert series of $S/J_C$ is \[H(S/J_C,t)=\frac{1}{(1-t)^{n+1}}((1+t)^{n-1}-t^{2}(1+t)^{n-1}+(n-1)t^{n}+t^{n+1}).\]
\end{enumerate}

\end{theorem}
 One might see the proof of the above Theorem in  \cite[Lemma 4.8 and Theorem 4.10]{So}.


In order to compute the Betti numbers of a cycle we need to understand the modules $J_L:g/J_L$ and $S/J_L:g$. To this end we need the following lemma about the canonical module of $S/J_L:g$.
\begin{lemma} \label{cc1} With the notation as before we have
\begin{itemize}
 \item[(a)] $\omega (S/J_L:g)\cong J_{\tilde{G}}/J_L(-2).$
 \item[(b)] The minimal number of generators of $\omega (S/J_L:g)$ is $\binom{n-1}{2}$.
\end{itemize}
\end{lemma}
\begin{proof} From Theorem \ref{c1} we have $J_L:g/J_L\cong \omega (S/J_{\tilde{G}})(2)$. Now consider the exact sequence
\[0\to \omega (S/J_{\tilde{G}})(2)\to S/J_L\to S/J_L:g\to 0.\]
All modules in above exact sequence are Cohen-Macaulay of dimension $n+1$. By applying local cohomology and dualizing it we get the following exact sequence
\[0\to \omega (S/J_L:g )\to S/J_L(-2)\to S/J_{\tilde{G}}(-2)\to 0.\] Which implies the isomorphism in (a) and then (a) gives us (b).
\end{proof}
All $\Tor$ modules of $J_L:g/J_L$ are given in the following lemma.
\begin{lemma} We have the following isomorphisms.
\begin{itemize}
 \item[(a)] $\Tor_i^S(K,J_L:g/J_L) \cong K^{c_i}(-n+2-i)$ for $i = 0,\ldots,n-2,$ where $c_i = (n-1-i) \binom{n}{i}$.
 \item[(b)] $\Tor_{n-1}^S(K,J_L:g/J_L) \cong K(-2n+2).$

\end{itemize}
\end{lemma}
\begin{proof} It is well known that $S/J_{\tilde{G}}$ is Cohen-Macaulay with the minimal free
resolution
\[0 \to S^{b_{n-1}}(-n) \to \cdots \to S^{b_{n-1-i}}(-n+i) \to \cdots \to S^{b_1}(-2) \to S\] where $b_i=i \binom{n}{i+1}$. By Lemma \ref{c1} (a) and Theorem \ref{p5}, we have the above statement.
\end{proof}
Now we will give the theorem in which we compute all $\Tor$ modules of $S/J_L:g.$
\begin{theorem} With the previous notation we have
\begin{itemize}
 \item[(a)] $\Tor_i^S(K,S/J_L:g) \cong K^{\binom{n-1}{i}}(-2i)\oplus K^{c_{i-1}}(-n+3-i)$ for $i = 1,\ldots,n-3,$
 \item[(b)] $\Tor_{n-2}^S(K,S/J_L:g) \cong K^{c_{n-3}}(-2n+5),$
 \item[(c)] $\Tor_{n-1}^S(K,S/J_L:g) \cong K^{\binom{n-1}{2}}(-2n+4),$
 \item[(d)] $\reg(S/J_L:g)=n-3.$
\end{itemize}
\end{theorem}
\begin{proof} Consider the exact sequence \[0\to J_L:g/J_L \to S/J_L\to S/J_L:g\to 0.\]
Let $i<n-2$, then the above exact sequence induces a graded homomorphism of degree zero
\[
\Tor_i^S(K,J_L:g/J_L)\cong K^{c_i}(-n+2-i) \to \Tor_i^S(K,S/J_L) \cong K^{\binom{n-1}{i}}(-2i).
\]
Therefore it is the zero homomorphism so we have the following isomorphism
\[
\Tor_i^S(K,S/J_L:g) \cong \Tor_i^S(K,S/J_L) \oplus \Tor_{i-1}^S(K,J_L:g/J_L).
\]
Let $i=n-1$, then we get the injection $0 \to K(-2n+2) \to K(-2n+2)$ which is in fact
an isomorphism. So we have the following exact sequence of K-vector spaces.
\begin{gather*}
0\to \Tor_{n-1}^S(K,S/J_L:g) \to \Tor_{n-2}^S(K,J_L:g/J_L)\cong K^{c_{n-2}}(-2n+4) \to  \Tor_{n-2}^S(K,S/J_L)\\ \cong K^{\binom{n-1}{1}}(-2n+4)\to\Tor_{n-2}^S(K,S/J_L:g) \to \Tor_{n-3}^S(K,J_L:g/J_L)\to 0.\end{gather*}
By Lemma \ref{cc1} (b) and Theorem \ref{p5} we have $\Tor_{n-1}^S(K,S/J_L:g) \cong K^{\binom{n-1}{2}}(-2n+4)$ since $S/J_L:g$ is Cohen-Macaulay. By investigating the K-vector space dimension of these modules and $c_{n-2}=\binom{n}{2}=\binom{n-1}{2}+\binom{n-1}{1}$, it follows that $\Tor_{n-2}^S(K,S/J_L:g) \cong \Tor_{n-3}^S(K,J_L:g/J_L).$
\end{proof}
\begin{lemma}\label{cc3} The coefficient of the highest power $t^{n-1}$ of the numerator of
the Hilbert series $H(S/J_C,t)$ is $\binom{n-1}{2}-1$.
\end{lemma}
\begin{proof} If we expand $(1+t)^{n-1}$ in the numerator of $H(S/J_C,t)$ of Theorem \ref{c1} (b). The last two terms in the numerator cancels and we get $1+(n-1)t+ \cdots +(\binom{n-1}{2}-1)t^{n-1}$.
\end{proof}
Now we are ready to say about all $\Tor$ modules of $S/J_C$.
\begin{theorem}\label{cc2} With the previous notation we have
\begin{itemize}
 \item[(a)] $\Tor_i^S(K,S/J_C) \cong K^{\binom{n}{i}}(-2i)\oplus K^{c_{i-2}}(-n+2-i)$ for $i = 1,\ldots,n-2,$
 \item[(b)] $\Tor_{n-1}^S(K,S/J_C) \cong K^{c_{n-3}}(-2n+3),$
 \item[(c)] $\Tor_{n}^S(K,S/J_C) \cong K^{\binom{n-1}{2}-1}(-2n+2).$

\end{itemize}
\end{theorem}
\begin{proof} Consider the exact sequence
\[0\rightarrow (S/J_L:g)(-2) \rightarrow S/J_L\rightarrow S/J_C\rightarrow 0.\]
Let $i<n-1$, then the above exact sequence induces a graded homomorphism of degree zero
\begin{gather*}\Tor_i^S(K,S/J_L:g)(-2)\cong K^{\binom{n-1}{i}}(-2i-2)\oplus K^{c_{i-1}}(-n+1-i) \\ \to \Tor_i^S(K,S/J_L) \cong K^{\binom{n-1}{i}}(-2i).\end{gather*}
First of all we observe that $\reg S/J_C = n-2$ as
Therefore it is the zero homomorphism so we have the following isomorphism
\[
\Tor_i^S(K,S/J_C) \cong \Tor_i^S(K,S/J_L) \oplus \Tor_{i-1}^S(K,S/J_L:g)(-2).
\]
Let $i=n$, then we have the following exact sequence of K-vector spaces
\begin{gather*}
0 \to \Tor_{n}^S(K,S/J_C) \to \Tor_{n-1}^S(K,S/J_L:g)(-2)\cong K^{\binom{n-1}{2}}(-2n+2) \to \Tor_{n-1}^S(K,S/J_L) \\ \cong K(-2n+2) \to  \Tor_{n-1}^S(K,S/J_C) \to \Tor_{n-2}^S(K,S/J_L:g)(-2) \cong K^{c_{n-3}}(-2n+3)\to 0.\end{gather*}
First of all we observe that $\reg S/J_C = n-2$ as
follows by the above isomorphisms and the previous
exact sequence. Moreover, the exact sequence
provides that $\Tor_n^S(K,S/J_C) \cong
K^{x}(-2n +2)$ for a certain positive integer
$x \in \mathbb{N}$. By view of the expression
of
\[ n - 2 = \reg{S/J_C} = \max \{ j-i \in \mathbb{Z} |
\beta_{i,j}(S/J_C) \not= 0\}
\]
it follows that $\beta_{i,j}(S/J_C) = 0$ for all $j - i > n-2$
and all $i > n$.
Therefore, computing the Hilbert series $H(S/J_C,t)$
by the minimal free resolution (see Lemma \ref{xyz}) has the form
\[H(S/J_C,t) = p(t)/(1-t)^{2n},\]
where $p(t)$ is a polynomial of degree $2n-2$
with leading term $(-1)^{n-1}\beta_{n,2n-2}(S/J_C)t^{2n-2}$.
Comparing it with the expansion of the Hilbert series as
given in Lemma \ref{cc3} it follows that
$\Tor_n^S(K,S/J_C) \cong K^{\binom{n-1}{2}-1}(-2n+2)$
which further implies that $\Tor_{n-1}^S(K,S/J_C) \cong
K^{c_{n-3}}(-2n+3)$.
\end{proof}
As a consequence of Theorem \ref{cc2} we are now able to describe the explicit Betti numbers of the binomial edge ideal of a cycle.

\begin{corollary}\label{cc4} Let $S/J_C$ be binomial edge ideal of cycle on vertex set $[n]$. Then the $\reg(S/J_C)=n-2$ and the Betti diagram of the $S/J_C$ looks like the following:
\[
\begin{array}{c|ccccccccc}

   & 0 & 1  & 2 &  \cdots  &  n-2  & n-1 & n  \\
\hline
0 & 1 & 0 & 0 & \cdots & 0 & 0 & 0 \\
1 & 0 & \beta_{1,2} & 0 & \cdots & 0 & 0 & 0   \\
2 & 0 & 0 & \beta_{2,4} & \cdots & 0 & 0 & 0  \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots
& \vdots \\
n-2 & 0 & 0 & \beta_{2,n} & \cdots  & \beta_{n-2,2n-4} & \beta_{n-1,2n-3}& \beta_{n,2n-2}
\end{array}
\]
where the Betti numbers in the diagonal are
$$\beta_{i,2i}=\binom{n}{i}\text{, if  } i=0,\ldots,n-3$$
 and the last row of Betti diagram is
\begin{eqnarray*}
\beta_{i,n-2+i}=\left\{\begin{array}{ll}
c_{i-2}, & \hbox{if \, $i = 2,\ldots,n-3$ ;}\\
\binom{n}{2}+ c_{n-4}, & \hbox{if \, $i=n-2$ ;}\\
 c_{n-3}, & \hbox{if \, $i=n-1$ ;}\\
 \binom{n-1}{2}-1, & \hbox{if \, $i=n.$}
\end{array}\right.
\end{eqnarray*}



\end{corollary}
\begin{proof} It follows from Theorem \ref{cc2}.
\end{proof}
\begin{corollary} \label{xx} Let G be any arbitrary graph on vertex set $[n]$. Let $C$ denote a cycle on maximal
$k$ vertices as an induced subgraph. Then $\reg(S/J_G)\geq k-2$ and $\beta_{i,j}(S/J_G) \geq \beta_{i,j}(S/J_C)$, where the values of $\beta_{i,j}(S/J_C)$ are those of Corollary \ref{cc4}
for $n = k$.
\end{corollary}
\begin{proof} It follows from Theorem \ref{p4} and Corollary \ref{cc4}.
\end{proof}

\begin{remark} In case $G$ has a cycle $C$ on maximal $k$ vertices as an induced subgraph it has also
a line $L$ on $k-1$ vertices. That is, the lower bound  $k-2 \leq \reg (S/J_G)$ is not better than that
of \cite{MM}. The advantage of Corollary \ref{xx} is that it provides the non-vanishing of certain
Betti numbers different from those of $\beta_{i,j}(S/J_L)$.

\end{remark}

\section{Betti Numbers of the binomial edge ideal of $\mathcal{T}_3$ and $\mathcal{G}_3$ }
The clique complex of a graph was used by many authors (see e.g. \cite{EHH},\cite{AR} and \cite{GR}) to study binomial edge ideals. Here in the following we introduce this nice concept.
\begin{definitions}Let $G$ be a simple graph on vertex set $[n]$.

\begin{enumerate}
                     \item A \textbf{clique} of $G$ is a subset $W$ of $[n]$
such that each vertex in $W$ is connected to any other vertex in $W$ by an edge
of $G$. In other words it is a complete subgraph of $G$.
\item A \textbf{maximal clique} is a clique that is not a subset of a larger clique.
                     \item The \textbf{clique complex} $\Delta(G)$ of $G$ is a simplicial complex whose facets are the maximal cliques of $G$.
                     \item A vertex $j\in[n]$ is called \textbf{free vertex} if it belongs to only one facet of $\Delta(G)$.
                   \end{enumerate}
\end{definitions}
For example, in a complete graph all vertices are free vertices, and in any graph the vertices of degree $1$ are free vertices, while cycle of length $> 3$ has no free vertices.

 The following proposition whose proof can be found  in \cite[Proposition 2.1]{AR} is important for us.
\begin{proposition}\label{acc} Let $G$ be a simple graph on vertex set $[n]$. Let $\Delta(G)$ is clique complex of $G$ and $j\in[n]$ be a vertex of $G$. Then the following conditions are equivalent:
\begin{enumerate}
                  \item $j$ is a free vertex of $\Delta(G)$.
                  \item $j \notin T$ for all $T\subseteq[n]$ such that $c(T \setminus \{i\})<c(T)$  for each $i \in T$.

                \end{enumerate}
\end{proposition}

The following lemma tells us the importance of the free vertex.
\begin{lemma}\label{b1}
 Let $G$ be the graph on vertex set $[n]$ with at least one free vertex and $J_{G}$ be its binomial edge ideal. Chose one of its free vertex and label it by $n$. Let $J_{G^\prime}$ denotes the binomial edge ideal of the graph $G^\prime$ by attaching the edge $\{n, n+1\}$ to the graph $G$. Then $f=x_ny_{n+1}-x_{n+1}y_n$ is regular on $S^\prime/J_{G}$ where $S^\prime=S[x_{n+1},y_{n+1}]$, and we have the following exact sequence of $S^\prime$-modules
 \[0\rightarrow S^\prime /J_{G}(-2)\mathop\rightarrow\limits^f S^\prime /J_{G}\rightarrow S^\prime /J_{G^\prime}\rightarrow 0.\]
where $J_{G^\prime}=(J_{G},f)$
\end{lemma}
\begin{proof}Since $n$ is free vertex therefore by Proposition \ref{acc} and Lemma \ref{p2} (c) $x_{n},y_{n}$ $\notin P_{T}(G)$
for all $\ P_{T}(G)\in \Ass(S/J_{G}),$ and hence $f$ $\notin P_{T}(G)$
for all $\ P_{T}(G)\in \Ass(S^\prime/J_{G})$. Therefore $f$ is not a zero divisor in $S^\prime/J_{G}$ and is regular, and one obtains the above exact sequence.
\end{proof}
\begin{definition}\label{d1} Let $\mathcal{T}_3$ be the collection of graphs such that for all $G\in\mathcal{T}_3$ we have
\[V (G) = \{u_1, \ldots , u_r, v_1, \ldots , v_s,w_1, \ldots ,w_t\}\]
with $r \geq 2, s \geq 1, t \geq 1$ and edge set

\begin{gather*}
E(G) =\{\{u_i, u_{i+1}\} : i = 1, \ldots , r - 1\} \cup \{\{v_i, v_{i+1}\} : i = 1, \ldots , s - 1\}\\ \cup
\{\{w_i,w_{i+1}\} : i = 1, \ldots , t - 1\} \cup \{\{u_1, v_1\}, \{u_1,w_1\}\}.
\end{gather*}
\end{definition}
Note that any $G\in\mathcal{T}_3$ is a tree which have at most one vertex of degree $3.$


\begin{example}\label{b2}

Consider the simplest example of the graph in $\mathcal{T}_3$ as shown in
Figure $1$. 

\[\begin{tikzpicture}
	
	\vertex[fill] (1) at (0,0) [label=below:$u_2$] {};
	\vertex[fill] (2) at (1.5,0) [label=below:$u_1$] {};
	\vertex[fill] (4) at (2.5,1) [label=above:$v_1$] {};
	\vertex[fill] (3) at (2.5,-1) [label=below:$w_1$] {};
	\path
		(1) edge (2)
		(3) edge (2)
		(2) edge (4)
		
		
		
	;
\end{tikzpicture}\]
\begin{center}
Figure \large{1}
\end{center}


It is easy to see that $u_2$, $v_1$ and $w_1$ are free vertices of $G$. Its binomial edge ideal has the following Betti diagram
\[
\begin{array}{c|cccc}
   & 0 & 1 & 2 &  3 \\
\hline
0 & 1 & 0 & 0  & 0 \\
1 & 0 & 3 & 0 &  0 \\
2 & 0 & 0 & 4 &  2
\end{array}
\]
and it has the following Hilbert series
\[H(S/J_{G},t)=\frac{1}{(1-t)^{6}}(1+2t-2t^{3}).\]

\end{example}
It is known from \cite[Corollary 3.5]{So} that any $G\in\mathcal{T}_3$ on the vertex set $[n]$, the ring $S/J_G$ is approximately Cohen-Macaulay ring of $\dim(S/J_G)=n+2$.
\begin{remark} We use computer algebra system CoCoA \cite{CO} for the computations of some arithmetic invariants of $S/J_G$ in Example \ref{b2} and \ref{b3}.
\end{remark}
\begin{definition}\label{d2} (see \cite{GR}) $\mathcal{G}_3$ be the collection of graphs such that for all $G\in\mathcal{G}_3$ we have
\[V (G) = \{u_1, \ldots , u_r, v_1, \ldots , v_s,w_1, \ldots ,w_t\}\]
with $r \geq 1, s \geq 1, t \geq 1$ and edge set
\begin{gather*}
E(G) =\{\{u_i, u_{i+1}\} : i = 1, \ldots , r - 1\} \cup \{\{v_i, v_{i+1}\} : i = 1, \dots , s - 1\}\\
\cup\{\{w_i,w_{i+1}\} : i = 1, \ldots , t - 1\} \cup \{\{u_1, v_1\}, \{u_1,w_1\}, \{v_1,w_1\}\}.
\end{gather*}

\end{definition}
\begin{example}\label{b3}

The simplest example of the graph in $\mathcal{G}_3$ is the complete graph on the vertex set $u_1$, $v_1$ and $w_1$ and all of them are free vertices of $G$. Its binomial edge ideal has the following Betti diagram
\[
\begin{array}{c|ccc}
   & 0 & 1 & 2  \\
\hline
0 & 1 & 0 & 0  \\
1 & 0 & 3 & 2

\end{array}
\]
and it has the following Hilbert series \[H(S/J_G,t)=\frac{1}{(1-t)^4}(1+2t).\]

\end{example}
It is known from \cite[Proposition 2.5]{GR} that any $G\in\mathcal{G}_3$ on vertex set $[n]$ is Cohen-Macaulay ring of $\dim(S/J_G)=n+1.$ Note that the $\projdim(S/J_G)=n-1$ for any $G\in\mathcal{T}_3\cup\mathcal{G}_3$ on vertex set $[n]$ as it will be shown in the following result.

\begin{theorem}\label{b4} Let $G$ be the graph on vertex set $[n]$ and $G\in\mathcal{T}_3\cup\mathcal{G}_3$. Let $J_G$ denotes its binomial edge ideal then the $\reg(S/J_G)=n-2$ and the Betti diagram of the $S/J_G$ looks like the following:

\[
\begin{array}{c|ccccccccc}

   & 0 & 1  & 2 &  3  &  4  & \cdots & n-2 & n-1  \\
\hline
0 & 1 & 0 & 0 & 0 & 0 & \cdots & 0& 0  \\
1 & 0 & \beta_{1,2} & \beta_{2,3} & 0 & 0 & \cdots & 0 & 0  \\
2 & 0 & 0 & \beta_{2,4} & \beta_{3,5} & 0 & \cdots & 0 & 0  \\
3 & 0 & 0 & 0 & \beta_{3,6} & \beta_{4,7} & \cdots & 0 & 0  \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots
& \vdots & \vdots\\
n-3 & 0 & 0 & 0 & 0& 0 & \ddots & \ddots & 0  \\
n-2 & 0 & 0 & 0 & 0 & 0 & \cdots & \beta_{n-2,2n-4}& \beta_{n-1,2n-3}
\end{array}
\]



\end{theorem}
\begin{proof}We want to prove the claim on the Betti table by induction on $n$. For $n=3$ or $n=4$ it is true (see Example \ref{b2} and \ref{b3}). Now  let us assume that the statement is true for $n$. We use the notations of Lemma \ref{b1} because in these classes we inductively go from graph $G$ on $n$ vertices to graph $G^\prime$ on $n+1$ vertices by adding an edge $\{n,n+1\}$ with the assumption that $n$ is free vertex. Let $F_{\bullet}$ be the minimal free resolution for $S/J_G$ then $F_{\bullet}\otimes_S S^\prime$ is minimal free resolution for $S^\prime /J_{G}$ and hence they have the same Betti numbers. To this end we note that
\[\Tor_i^S(K,S /J_{G})\cong \Tor_i^{S^\prime}(K,S^\prime /J_{G})\]
Hence we have the following isomorphism restricted to degree $i+j$.
\[\Tor_i^{S^\prime}(K,S^\prime /J_{G})_{i+j}\cong \Tor_i^S(K,S /J_{G})_{i+j} \cong K^{\beta_{i,i+j}}(-(i+j)) \]
Now consider the exact sequence of Lemma \ref{b1}
\[0\rightarrow S^\prime /J_{G}(-2)\mathop\rightarrow\limits^f S^\prime /J_{G}\rightarrow S^\prime /J_{G^\prime}\rightarrow 0.\]
The first two modules of the above exact sequence are the same modules with the shift of degree $2$. Next we want to show that the map
\[
\phi_{i}: \Tor_i^{S^\prime}(K,S^\prime /J_{G}(-2)) \to \Tor_i^{S^\prime}(K,S^\prime /J_{G}).
\]
is the zero map. To this end it will be enough to show that
\[
[\phi_{i}]_{j}: \Tor_i^{S^\prime}(K,S^\prime /J_{G})_{i+j-2} \to \Tor_i^{S^\prime}(K,S^\prime /J_{G})_{i+j}.
\]
is zero for all $j$. Now suppose that
\[
0 \neq \Tor_i^{S^\prime}(K,S^\prime /J_{G})_{i+j-2} \cong \Tor_i^{S}(K,S /J_{G})_{i+j-2}.
\]
By induction hypothesis for $n$ it turns out that $(i,j-2)$ is either $(i,i)$ or $(i,i-1)$. In the first case, that is $j-2=i$, the target of $[\phi_{i}]_{j}$ is
\[
\Tor_i^{S^\prime}(K,S^\prime /J_{G})_{2i+2} \cong \Tor_i^{S}(K,S /J_{G})_{2i+2}=0.
\]
In the second case, that is $j-2=i-1$, the target of $[\phi_{i}]_{j}$ is
\[
\Tor_i^{S^\prime}(K,S^\prime /J_{G})_{2i+1}\cong \Tor_i^{S}(K,S /J_{G})_{2i+1}=0.
\]
Now suppose that the target of $[\phi_{i}]_{j}$ namely $\Tor_i^{S^\prime}(K,S^\prime /J_{G})_{i+j}$ is non-zero. Again by induction hypothesis for $n$ it follows that $(i,j)$ is either $(i,i)$ or $(i,i-1)$. In the first case, that is $j=i$, the source of $[\phi_{i}]_{j}$ is
\[
\Tor_i^{S^\prime}(K,S^\prime /J_{G})_{2i-2} \cong \Tor_i^{S}(K,S /J_{G})_{2i-2}=0.
\]
In the second case, that is $j=i-1$, the source of $[\phi_{i}]_{j}$ is
\[
\Tor_i^{S^\prime}(K,S^\prime /J_{G})_{2i-3} \cong \Tor_i^{S}(K,S /J_{G})_{2i-3}=0.
\]
This completes the proof for $\phi_{i}$ is the zero map. Therefore the short exact sequence induces an isomorphism
\[
\Tor_i^{S^\prime}(K,S^\prime /J_{G^\prime}) \cong \Tor_i^{S^\prime}(K,S^\prime /J_{G}) \oplus \Tor_{i-1}^{S^\prime}(K,S^\prime /J_{G}(-2)).
\]
In order to complete the inductive step we have to show that $\Tor_i^{S^\prime}(K,S^\prime /J_{G^\prime})_{i+j}$ is zero for all $(i,j)$ different of $(i,i)$ and $(i,i-1)$. This follows because of
\[
\Tor_i^{S^\prime}(K,S^\prime /J_{G^\prime})_{i+j} \cong \Tor_i^{S^\prime}(K,S^\prime /J_{G})_{i+j} \oplus \Tor_{i-1}^{S^\prime}(K,S^\prime /J_{G})_{i+j-2}.
\]
Note that if $(i,j)\neq(i,i-1)$ and $(i,j)\neq(i,i)$, then $(i-1,j-1)\neq(i-1,i-2)$ and $(i-1,j-1)\neq(i-1,i-1)$.
Moreover we get
\[
\beta_{i,2i}(S^\prime /J_{G^\prime}) = \beta_{i,2i}(S/J_{G})+ \beta_{i-1,2i-2}(S/J_{G}).
\]
and
\[
\beta_{i,2i-1}(S^\prime /J_{G^\prime}) = \beta_{i,2i-1}(S/J_{G})+ \beta_{i-1,2i-3}(S/J_{G}).
\]
Hence any Betti number of $S^\prime /J_{G^\prime}$ is the sum of consecutive Betti numbers of $S/J_{G}$ of its same diagonal. This completes the inductive step.
\end{proof}
The recursion formulas for the Betti numbers at the end of the proof might be used for an explicit computation of them. We will follow here a different approach using Hilbert series.
\begin{lemma}\label{b5}
    With the notations of Lemma \ref{b1}, we have $$H(S^\prime /J_{G^\prime},t)=(1-t^{2})H(S^\prime /J_{G},t).$$
\end{lemma}
\begin{proof}
The desired identity of Hilbert series
follows from the exact sequence
\[0\rightarrow S^\prime /J_{G}(-2)\mathop\rightarrow\limits^f S^\prime /J_{G}\rightarrow S^\prime /J_{G^\prime}\rightarrow 0\] by using the additivity of Hilbert series on short exact sequences.
\end{proof}
\begin{lemma}\label{b6} Let $G$ be a graph on vertex set $[n]$ and $J_G$ be its binomial edge ideal.
\begin{enumerate}
  \item[(a)] Let $G\in\mathcal{T}_3$ then the Hilbert series is
  \[ H(S/J_{G},t)=\frac{1}{(1-t)^{n+2}}(1+2t-2t^{3})(1+t)^{n-4}\text{ \ for }n>3.\]
  \item[(b)] Let $G\in\mathcal{G}_3$ then the Hilbert series is
  \[ H(S/J_{G},t)=\frac{1}{(1-t)^{n+1}}(1+2t)(1+t)^{n-3}\text{ \ for }n>2.\]
\end{enumerate}
\end{lemma}
\begin{proof}
We will prove (a) by induction on $n$. For $n=4$ it is true, see Example \ref{b2}. Suppose the claim is true for $n.$ That is,
 $$H(S^\prime /J_{G},t)=\frac{1}{(1-t)^{n+4}}(1+2t-2t^{3})(1+t)^{n-4}.$$ Now by previous lemma we have
\begin{equation*}
H(S^\prime /J_{G^\prime},t)=\frac{1}{(1-t)^{n+3}}(1+2t-2t^{3})(1+t)^{n-3}
\end{equation*}%
as required.

Similar arguments might be used in order to calculate the Hilbert series
in (b).
\end{proof}
\begin{theorem}\label{b7} Let $G$ be a graph on vertex set $[n]$ and $J_G$ be its binomial edge ideal.
\begin{enumerate}
  \item[(a)] Let $G\in\mathcal{T}_3$ then the Betti numbers for $S/J_G$ are:
  \begin{eqnarray*}
\beta_{i,j}=\left\{\begin{array}{ll}
\binom{n-4}{i}+3\binom{n-4}{i-1}+4\binom{n-4}{i-2}, & \hbox{if \, $j=2i$ and $i=0,\ldots,n-2$ ;}\\
2\binom{n-4}{i-3}, & \hbox{if \, $j=2i-1$ and $i=3,\ldots,n-1$ ;}\\
 0, & \hbox{ \, otherwise .}

\end{array}\right.
\end{eqnarray*}
  \item[(b)] Let $G\in\mathcal{G}_3$ then the Betti numbers for $S/J_G$ are:
  \begin{eqnarray*}
\beta_{i,j}=\left\{\begin{array}{ll}
3\binom{n-3}{i-1}+\binom{n-3}{i}, & \hbox{if \, $j=2i$ and $i=0,\ldots,n-2$ ;}\\
2\binom{n-3}{i-2}, & \hbox{if \, $j=2i-1$ and $i=2,\ldots,n-1$ ;}\\
 0, & \hbox{ \, otherwise .}

\end{array}\right.
\end{eqnarray*}
  \end{enumerate}
\end{theorem}
\begin{proof}
By Lemma \ref{xyz} and the structure of the Betti table (see in the above Theorem \ref{b4}) provides the following formula
\[
H(S/J_G,t) = \frac{1}{(1-t)^{2n}}\left(\sum_{i=0}^{n-2} (-1)^i\beta_{i,2i} t^{2i}+\sum_{i=2}^{n-1} (-1)^{i}\beta_{i,2i-1} t^{2i-1}\right).
\]
and after comparing the Hilbert series of Lemma \ref{b6} (a) and by making a few simple computations we have the formulas for the Betti numbers in (a). Similar computation using the Hilbert series of Lemma \ref{b6} (b) gives the Betti numbers in (b).
\end{proof}
\begin{corollary}\label{xy} Let G be any arbitrary graph on vertex set $[n]$. Suppose that $G$ has an induced subgraph $H\in\mathcal{T}_3\cup\mathcal{G}_3$ on $k$ vertices. Then $\reg(S/J_G)\geq k-2$.
\end{corollary}
\begin{proof} It is an easy consequence of Theorem \ref{p4} and \ref{b4}.
\end{proof}

\begin{remark}  Let $G$ denote a graph with the largest $k$ such that $G$ has an induced
subgraph $H \in\mathcal{T}_3\cup\mathcal{G}_3$. Then it has also a line $L$ as an induced subgraph
with $\ell = \max \{s+t+1,r+t,r+s\}$ resp. $\ell = \max \{r+s,r+t,s+t\}$ vertices.
In general $k > \ell$, so
that the lower bound for the Castelnuovo-Mumford regularity in Corollary \ref{xy} improves those of \cite{MM}.
\end{remark}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Acknowledgements:}
The authors are grateful to the reviewer for suggestions to improve
the presentation of the manuscript.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% If you use BibTeX to create a bibliography
% then copy and past the contents of your .bbl file into your .tex file

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