Related papers: Edge ideals whose all matching powers are bi-Cohen…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. For all $s \geq 1$, we obtain upper bounds for reg$(I(G)^s)$ for bipartite graphs. We then compare the properties of $G$ and $G'$, where $G'$ is the graph…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field $K$, and let $A$ be a finitely generated standard graded $S$-algebra. We show that if the defining ideal of $A$ has a quadratic initial ideal, then all the graded components of…

Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this…

We first characterise graphs with binomial edge ideals of K\"onig type as those for which the path covering number is equal to a minor variant of the scattering number. These are well-studied graph-theoretic invariants, allowing us to apply…

In this paper we prove the conjectured upper bound for Castelnuovo-Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the…

We classify all complete uniform multipartite hypergraphs with respect to some algebraic properties, such as being (almost) complete intersection, Gorenstein, level, $l$-Cohen-Macaulay, $l$-Buchsbaum, unmixed, and satisfying Serre's…

We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a lexicographic order induced by a vertex…

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose…

The study of the edge ideal $I(D_{G})$ of a weighted oriented graph $D_{G}$ with underlying graph $G$ started in the context of Reed-Muller type codes. We generalize a Cohen-Macaulay construction for $I(D_{G})$, which Villarreal gave for…

Assume that $G$ is a graph with edge ideal $I(G)$ and matching number ${\rm match}(G)$. For every integer $s\geq 1$, we denote the $s$-th squarefree power of $I(G)$ by $I(G)^{[s]}$. It is shown that for every positive integer $s\leq {\rm…

We discuss algebraic and homological properties of binomial edge ideals associated to graphs which are obtained by gluing of subgraphs and the formation of cones.

Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2 \rfloor}, and let I(G) denote its the edge ideal in the ring R = k[x_1,...,x_n]. We consider the problem of determining when G is Cohen-Macaulay, i.e, R/I(G) is a…

We graph-theoretically characterize the class of graphs $G$ such that $I(G)^2$ are Buchsbaum.

We classify all normal edge ideals of edge-weighted graphs.

Monomial ideals corresponding to strong quasi-n-partite graphs are considered. Some algebraic and combinatorial properties of generalized graph ideals of a strong quasi-n-partite graph are studied. Furthermore, we show that the edge ideal…

We describe combinatorially the Cohen-Macaulay type of edge-weighted r-path suspensions of edge-weighted graphs for an arbitrary positive integer r. The computation of the Cohen-Macaulay type of edge-weighted suspensions of edge-weighted…

For a graph $G$, Bolognini et al. have shown $J_{G}$ is strongly unmixed $\Rightarrow$ $J_{G}$ is Cohen-Macaulay $\Rightarrow$ $G$ is accessible, where $J_{G}$ denotes the binomial edge ideals of $G$. Accessible and strongly unmixed…

A graded ideal $I$ in $\mathbb{K}[x_1,\ldots,x_n]$, where $\mathbb{K}$ is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree $\mathrm{pd}(I)-2$, while it is not linear…

We classify connected graphs $G$ whose binomial edge ideal is Gorenstein. The proof uses methods in prime characteristic.

Let $\operatorname{pd}(I(G))$ and $\operatorname{reg}(I(G))$ respectively denote the projective dimension and the regularity of the edge ideal $I(G)$ of a graph $G$. For any positive integer $n$, we determine all pairs…