Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.

In this paper, we explore the relation between index of reducibility and Hilbert coefficients in local rings. Consequently, main result study provides a characterization sequentially Cohen-Macaulay ring terms its non-parameter ideals. As corollaries to theorem, obtain characterizations Gorenstein/Cohen-Macaulay Chern

an isomorphism? These questions have interest partly because if S and .S’(J, S) are Cohen-Macaulay, then so is gr, S := S/J@ J/J’... [ 16 1 and under these hypotheses if N is perfect and R @ .S(J, S) = .$(I, R), then R and .7(1. R) are Cohen-Macaulay too. Thus, gr,R is Cohen-Macaulay, and its torsion freeness and normality, for exmple, can be characterized in terms of analytic spreads, as in [ ...

We describe a combinatorial condition on graphwhich guarantees that all powers of its vertex cover ideal are componentwise linear. Then motivated by Eagon and Reiner’s Theorem we study whether the Cohen-Macaulay graph have linear free resolutions. After giving complete characterization cactus graphs (i.e., connected in which each edge belongs to at most one cycle) show their ideals

We prove that a Cohen-Macaulay normal variety X has Du Bois singularities if and only if π∗ωX′(G) ≃ ωX for a log resolution π : X ′ → X , where G is the reduced exceptional divisor of π. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward an...

Let G be a simple graph on n vertices and JG denote the binomial edge ideal of in polynomial ring S=K[x1,…,xn,y1,…,yn]. In this article, we compute second graded Betti numbers JG, obtain minimal presentation it when is tree or unicyclic graph. We classify all graphs whose ideals are almost complete intersection, prove that they generated by d-sequence Rees algebra their Cohen-Macaulay. also an ...

We introduce a new family of simple graphs, so called, growing graphs. investigate ways to modify given graph G combinatorially obtain graph. One may infinitely many graphs from single show that obtained any is Cohen–Macaulay and every chordal also prove under certain conditions, if only its clique complex grafted give several equivalent conditions in this case. Our work inspired by generalizes...

In a recent paper, E. Steingŕımsson associated to each simple graph G a simplicial complex ∆G denoted as the coloring complex of G. Certain nonfaces of ∆G correspond in a natural manner to proper colorings of G. Indeed, the h-vector is an affine transformation of the chromatic polynomial χG of G, and the reduced Euler characteristic is, up to sign, equal to |χG(−1)| − 1. We show that ∆G is cons...