Depth and Extremal Betti Number of Binomial Edge Ideals

Let $G$ be a simple graph on the vertex set $[n]$ and $J_G$ be the corresponding binomial edge ideal. Let $G=v\ast H$ 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 $S\mathrm{/}{J}_G$ in terms of Cohen-Macaulay defect of $S_H\mathrm{/}{J}_H$ and using this we construct a graph with Cohen-Macaulay defect $q$ for any $q\ge 1$. 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 $1\le b<r$, there exists a connected graph $G$ such that $reg(S\mathrm{/}{J}_G)=r$ and the number of extremal Betti number of $S\mathrm{/}{J}_G$ is $b$.