prove the following result.

\[main2\] Let $G$ be a simple graph on the vertex set $[n]$ with no isolated vertices, and let $I$ be its edge ideal. The following conditions are equivalent:

-   The complement $\bar{G}$ of $G$ is a cycle of length $n$.

-   $\index(I)=\projdim(I)$.

If the equivalent conditions hold, then $\projdim(I)=n-3$.

To prove this theorem we need some intermediate steps. We first observe the following fact which will used several times in the sequel:

Let $G$ be a graph on the vertex set $[n]$ and $\Delta(G)$ be its clique complex. For any $W\subseteq [n]$ we have $$\begin{aligned}
\label{Delta1}
\Delta(G_W)=\Delta(G)_W.\end{aligned}$$ In other words, $\Delta(G)_W$ is the clique complex of induced subgraph of $G$ on the vertex set $W$. Moreover, $G_W$ is connected if and only if $\Delta_W$ is connected. Here $G_W$ (resp. $\Delta(G)_W$) denotes the induced subgraph of $G$ (resp. the induced subcomplex of $\Delta(G)$) whose vertex set is $W$.

\[index\] Let $G$ be a simple graph on the vertex set $[n]$ and $I$ its edge ideal. Suppose that $\index(I)=\projdim(I)=t$. Then $\beta_{t,t+2}(I)=0$

By using Hochster’s formula [@HH Theorem 8.1.1], it is enough to show that $\widetilde{H}_{0}(\Delta_{W}; K)= 0$ for any $W\subset [n]$ with $|W|=t+2$. To show this, it is 