Related papers: Homological shifts of powers a complementary edge …
For a monomial ideal $I$, we consider the $i$th homological shift ideal of $I$, denoted by $\text{HS}_i(I)$, that is, the ideal generated by the $i$th multigraded shifts of $I$. Some algebraic properties of this ideal are studied. It is…
Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. In this paper, we introduce the $i$th \textit{homological shift algebras}…
Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ and $I\subset S$ be a squarefree monomial ideal generated in degree $n-2$. Motivated by the remarkable behavior of the powers of $I$ when $I$ admits a linear resolution, as…
We determine all possible triples of depth, dimension, and regularity of edge ideals of weighted oriented graphs with a fixed number of vertices. Also, we compute all the possible Betti table sizes of edge ideals of weighted oriented trees…
Let $I$ be a polymatroidal ideal. In this paper, we study the asymptotic behavior of the homological shift ideals of powers of polymatroidal ideals. We prove that the first homological shift algebra $\text{HS}_1(\mathcal{R}(I))$ of $I$ is…
A monomial ideal $I$ is said to have homological linear quotients if for each $k\geq 0$, the homological shift ideal $\mathrm{HS}_k(I)$ has linear quotients. It is a well-known fact that if an edge ideal $I(G)$ has homological linear…
Let $G$ be a finite simple graph, and let $I(G)$ denote its edge ideal. In this paper, we investigate the asymptotic behavior of the syzygies of powers of edge ideals through the lens of homological shift ideals $\text{HS}_i(I(G)^k)$. We…
In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to vertex weighted rooted forests and oriented cycles. As some consequences, we give some exact formulas for the depth of…
Given a square-free monomial ideal $I$, satisfying certain hypotheses, in a polynomial ring $R$ over a field $\mathbb{K}$, we compute the projective dimension of $I$. Specifically, we focus on the cases where the 1-skeleton of an associated…
In this paper, we investigate the degree of $h$-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of the $h$-polynomial for various fundamental classes of graphs such as…
Let $G$ be a connected and simple graph on the vertex set $[n]$. To the graph $G$ one can associate the generalized binomial edge ideal $J_{m}(G)$ in the polynomial ring $R=K[x_{ij}: i \in [m], j \in [n]]$. We provide a lower bound for the…
In this paper, we introduce the concept of complementary edge ideals of graphs and study their algebraic properties and invariants.
We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the…
In this article, we give a comprehensive survey of the recent progress of research on binomial edge ideal of a graph since 2018.
In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to three special types of vertex-weighted oriented $m$-partite graphs. These formulas are functions of the weight and…
In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals of vertex-weighted oriented unicyclic graphs. These formulas are in function of the weight of the vertices, the numbers of edges. We…
Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is…
In this article we study higher homological properties of $n$-levelled algebras and connect them to properties of the underlying graphs. Notably, to each $2$-representation-finite quadratic monomial algebra $\Lambda$ we associate a…
Edge ideals of finite simple graphs $G$ on $n$ vertices are the ideals $I(G)$ of the polynomial ring $S$ in $n$ variables generated by the quadratic monomials associated with the edges of $G$. In this paper, we consider the possible pairs…
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…