Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $G=v*H$ 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 $b$.

29
The categories of this paper are: funct-an
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex