Depth and extremal Betti number of binomial edge ideals

Let G be a simple graph on the vertex set [ n ] and let 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 the Betti numbers of J H and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen–Macaulay defect of S / J Gin terms of Cohen–Macaulay defect ofS H / J Hand using this we construct a graph with Cohen–Macaulay defect q for any q ≥ 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 ≤ b < r , there exists a connected graph G such that reg ( S / J G ) = r and the number of extremal Betti numbers of S / J Gis b .