Related papers: Regularity of powers of forests and cycles

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper we prove that if $G$ is a unicyclic graph then for all $s \geq 1$ the regularity of $I(G)^s$ is exactly $2s+\text{reg}(I(G))-2$. We also…

Let $I = I(G)$ be the edge ideal of a graph $G$. We give various general upper bounds for the regularity function $\text{reg} I^s$, for $s \ge 1$, addressing a conjecture made by the authors and Alilooee. When $G$ is a gap-free graph and…

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 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 $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive…

Let $L_n$ be a line graph with $n$ vertices and let $I$ be its $t$-path ideal. It is shown that $I^s$ has a linear resolution for some $s\geq 1$ (or equivalently for all $s\geq 1$) if and only if $I^s$ has linear quotients for some $s\geq…

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 $q \geq 1,$…

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 $G$ 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…

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

Let $G$ be a simple graph and $I_3(G)$ be its $3$-path ideal in the corresponding polynomial ring $R$. In this article, we prove that for an arbitrary graph $G$, $reg(R/I_3(G))$ is bounded below by $2\nu_3(G)$, where $\nu_3(G)$ denotes the…

Let $G=W(C_{n})$ be a whiskered cycle graph with edge ideal $I=I(G)$. We prove that for every $s\geq 1$, the equality ${\rm reg}(I^{s})=2s+\lceil \frac{n-1}{2}\rceil-1$ holds.

In this paper we prove that if $I(G)$ is a bipartite edge ideal with regularity three then for all $s\geq 2$ the regularity of $I(G)^s$ is exactly $2s+1$.

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper we prove that if $G$ is a very well-covered graph then for all $s \geq 2$ the regularity of $I(G)^s$ is exactly $2s+\nu(G)-1$.

Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${\rm reg}(I^{(s)}) = {\rm reg}(I^s)$$ for $s = 2,3$, where $I^{(s)}$ is the $s$-th symbolic power of $I$. As a consequence, we prove the following bounds \begin{align*} {\rm…

Let $I(G)$ be the edge ideal of a simple graph $G$ over a field k. We prove that $${\rm reg}(\overline {I(G)^s}) = {\rm reg}(I(G)^s),$$ for all $s \le 4$. Furthermore, we provide an example of a graph $G$ such that $${\rm reg} I(G)^s = {\rm…

Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G…

Assume that $G$ is a graph with edge ideal $I(G)$ and let $I(G)^{(s)}$ denote the $s$-th symbolic power of $I(G)$. It is proved that for every integer $s\geq 1$, $${\rm reg}(I(G)^{(s+1)})\leq \max\bigg\{{\rm reg}(I(G))+2s, {\rm…

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…