Related papers: Normally torsion-free edge ideals of weighted orie…

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. If $D$ contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary and sufficient condition…

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…

For the edge ideal $I(\D)$ of a weighted oriented graph $\D$, we prove that its symbolic powers $I(\D)^{(t)}$ are Cohen-Macaulay for all $t\geqslant 1$ if and only if the underlying graph $G$ is composed of a disjoint union of some complete…

Let $D$ be a weighted oriented graph with the underlying graph $G$ when vertices with non-trivial weights are sinks and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G,$ respectively. We give explicit description of the…

Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial…

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(\mathcal{D})$ coincide when $\mathcal{D}$ is a weighted oriented certain class of…

In this paper, we study the componentwise linearity of powers of edge ideal of a weighted oriented graph $D$. We give a characterization for componentwise linearity of the edge ideal $I(D)$ in terms of forbidden subgraphs of $D$. If $D$ is…

Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $\mathcal{D}$ contains a perfect matching consisting of leaves, we provide…

In this paper, we describe primary decomposition of the edge ideal of the join of some graphs in terms of that information of the edge ideal of every weighted oriented graph. Meanwhile, we also study depth and regularity of symbolic powers…

In this paper, we focus on the associated primes of powers of monomial ideals and asymptotic behavior properties such as normally torsion-freeness, normality, the strong persistence property, and the persistence property. In particular, we…

An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all natural numbers t. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a…

Let $D$ be a weighted oriented graph with the underlying graph $G$ and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G$ respectively. We show that the regularity of edge ideal of a certain class of weighted oriented graph…

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with…

Let $I(G)$ be the edge ideal of a simple graph $G$. We prove that $I(G)^2$ has a linear free resolution if and only if $G$ is gap-free and reg$I(G) \le 3$. Similarly, we show that $I(G)^3$ has a linear free resolution if and only if $G$ is…

In this paper, we first give some sufficient criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property.…

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…

Let (P,<) be a finite poset and let G be its comparability graph. If cl(G) is the clutter of maximal cliques of G, we prove that cl(G) satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. We prove that…

In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if $D$ is a weighted oriented graph whose edge ideal $I(D)$ is componentwise linear, then the underlying simple graph $G$ of $D$ is…

In this paper, our main focus is to explore different classes of nearly normally torsion-free ideals. We first characterize all finite simple connected graphs with nearly normally torsion-free cover ideals. Next, we characterize all…

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which…