# 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.

## Key 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 $b$.