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Let G be a finite simple graph on n vertices and J G denote the corresponding binomial edge ideal in the polynomial ring S = K[x 1 , . . . , x n , y 1 , . . . , y n ]. In this article, we compute the Hilbert series of binomial edge ideal of decomposable graphs in terms of Hilbert series of its indecomposable subgraphs. Also, we compute the Hilbert series of binomial edge ideal of join of two graphs and as a consequence we obtain the Hilbert series of complete k-partite graph, fan graph, multi-fan graph and wheel graph.

Let G be a simple graph on n vertices and J G denote the corresponding binomial edge ideal in the polynomial ring S = K[x 1 , . . . , x n , y 1 , . . . , y n ]. In this article, we compute the second Betti number and obtain a minimal presentation of trees and unicyclic graphs. We also classify all graphs whose binomial edge ideals are almost complete intersection and we prove that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals.

Let G be a simple graph on the vertex set [n] and let JG 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 JG in terms of the Betti numbers of JH and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen–Macaulay defect of S/JG in terms of Cohen–Macaulay defect of SH/JH and 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

Let S = k[x 1 , . . . , x n ] be a polynomial ring, where k is a field, and G be a simple graph on n vertices. Let J(G) ⊂ S be the cover ideal of G. In this article, we solve a conjecture due to Herzog, Hibi and Ohsugi for trees which states that powers of cover ideals of trees are componentwise linear. Also, we show that if G is a unicyclic vertex decomposable graph unless it contains C 3 or C 5 , then symbolic powers of J(G) are componentwise linear.