Related papers: Regularity of Powers and symbolic powers of edge i…

In this article, we prove that for several classes of graphs, the Castelnuovo-Mumford regularity of symbolic powers of their edge ideals coincide with that of their ordinary powers.

In this paper, we compute the regularity and Hilbert series of symbolic powers of the cover ideal of a graph $G$ when $G$ is either a crown graph or a complete multipartite graph. We also compute the multiplicity of symbolic powers of cover…

In this paper we provide some exact formulas for the regularity of powers of edge ideals of vertex-weighted oriented cycles and vertex-weighted unicyclic graphs. These formulas are functions of the weight of vertices and the number of…

We study the regularity of small symbolic powers and integral closure of small powers of edge ideals. We also prove that the regularity of integral closure of powers of edge ideals of graphs with at most two odd cycles is the same as the…

We compute the Castelnuovo-Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and…

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 study the componentwise linearity of symbolic powers of edge ideals. We propose the conjecture that all symbolic powers of the edge ideal of a cochordal graph are componentwise linear. This conjecture is verified for some…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of symbolic powers of certain…

Let $G$ be a graph and let $I = I(G)$ be its edge ideal. When $G$ is unicyclic, we give a decomposition of symbolic powers of $I$ in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the…

Let $G$ be a unicyclic graph with edge ideal $I(G)$. For any integer $s\geq 1$, we denote the $s$-th symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that ${\rm reg}(I(G)^{(s)})={\rm reg}(I(G)^s)$, for every $s\geq 1$.

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{reg} I(G)$ and the asymptotic linear function $\text{reg} I(G)^q$, for $q \geq 1,$…

Let $G$ be a simple graph and $I(G)$ be its edge ideal. In this article, we study the Castelnuovo-Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh's conjecture for wheel graphs,…

We prove that the regularity of edge ideals of powers of forests is weakly decreasing. We then compute the regularity of edge ideals of powers of cycles.

Let $G$ be a simple graph. We demonstrate a method for using $t$-admissible subgraphs of $G$ to determine the regularity of the $t$-th symbolic power of the cover ideal of $G$. As an application, we compute the regularity of powers of cover…

Let $G$ be a graph and $I=I(G)$ be its edge ideal. When $G$ is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of $I$ and compute the Waldschmidt…

This paper presents exact formulas for the regularity and depth of powers of edge ideals of an edge-weighted star graph. Additionally, we provide exact formulas for the regularity of powers of the edge ideal of an edge-weighted integrally…

This paper gives exact formulas for the regularity of powers of edge ideals of an edge-weighted integrally closed cycle.

Let $G_{n,r}$ denote the graph with $n$ vertices $\{x_1,\ldots,x_n\}$ in cyclic order and for each vertex $x_i$ consider the set $A_i=\{x_{i-r},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots, x_{i+r}\},$ where $x_{i-j}$ is the vertex $x_{n+i-j}$,…

For a finite simple graph $G$ we give an upper bound for the regularity of the powers of the edge ideal $I(G)$.

In this paper, we compare the regularities of symbolic and ordinary powers of edge ideals of weighted oriented graphs. For any weighted oriented complete graph $K_n$, we show that $\reg(I(K_n)^{(k)})\leq \reg(I(K_n)^k)$ for all $k\geq 1$.…