Related papers: Edge ideals of some edge-weighted graphs

Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity of R/I(C) and apply this criterion to give…

The regularity of an edge ideal of a finite simple graph $G$ is at least the induced matching number of $G$ and is at most the minimum matching number of $G$. If $G$ possesses a dominating inuduced matching, i.e., an induced matching which…

In this paper it is shown that it is possible to associate several polynomial ideals to a directed graph $D$ in order to find properties of it. In fact by using algebraic tools it is possible to give appropriate procedures for automatic…

Let $I$ and $J$ be edge ideals in a polynomial ring $R = \mathbb{K}[x_1,\ldots,x_n]$ with $I \subseteq J$. In this paper, we obtain a general upper and lower bound for the Castelnuovo-Mumford regularity of $IJ$ in terms of certain…

We compute the multigraded Betti numbers of edge ideals for weighted oriented crown graphs using a novel approach. This approach, which we still call the \emph{induced subgraph approach}, originates from our prior work on computing Betti…

In this paper, we obtain a combinatorial formula for computing the Betti numbers in the linear strand of edge ideals of bipartite Kneser graphs. We deduce lower and upper bounds for regularity of powers of edge ideals of these graphs in…

In this paper we study the $s$-th symbolic powers of the edge ideals of complete graphs. In particular, we provide a criterion for finding an Eliahou-Kervaire splitting on these ideals, and use the splitting to provide a description for the…

Let $G$ be a finite graph on $[n]:=\{1, \ldots, n\}$ and $\kappa(G)$ its vertex connectivity. Let $S=K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G^c)$ the edge ideal of the complementary graph…

Assume that $G$ is a graph with edge ideal $I(G)$. For every integer $s\geq 1$, we denote the squarefree part of the $s$-th symbolic power of $I(G)$ by $I(G)^{\{s\}}$. We determine an upper bound for the regularity of $I(G)^{\{s\}}$ when…

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone…

For any two integers $d,r \geq 1$, we show that there exists an edge ideal $I(G)$ such that the ${\rm reg}\left(R/I(G)\right)$, the Castelnuovo-Mumford regularity of $R/I(G)$, is $r$, and ${\rm deg} (h_{R/I(G)}(t))$, the degree of the…

We determine the arithmetical rank of every edge ideal of a Ferrers graph.

In this article we obtain an improved upper bound for the regularity of binomial edge ideals of trees.

In this article, we give a comprehensive survey of the recent progress of research on binomial edge ideal of a graph since 2018.

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. We provide one method to find all the minimal generators of $ I_{\subseteq C} $, where $ C $ is a maximal strong vertex cover of $D$ and $ I_{\subseteq C} $ is the…

Combinatorial properties of some ideals related to strong quasi-n-partites graphs are examined. We prove that the edge ideal of a strong quasi-n-partite graph is not integrally closed and we give an expression for its integral closure.…

In this article, we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique. As a consequence, we obtain an upper bound for the regularity of binomial edge ideal of a…

Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal. The aim of this note is to investigate different aspects of the Rees algebra $\mathcal{R}(I)$ of $I$. We compute its regularity and the universal Gr\"obner basis of its defining…

We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of line graphs and cycles. As a consequence we can give new and short proofs for the known formulas of regularity and projective…

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then we show that for the edge ideal $I(G)$ of a graph $G$ the induced matching number of $G$ is an upper bound for the v-number of $I(G)$ when…