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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 reg} I^{s} & le {rm reg} I + 2s - 2, text{ for } s = 2,3, {rm reg} I^{(s)} & le {rm reg} I + 2s - 2,text{ for } s = 2,3,4. end{align*}
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.
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 tree. Fi
In this paper, we define (im, reg)-invariant extension of graphs and propose a new approach for Nevo and Peevas conjecture which said that for any gap-free graph $G$ with $reg(I(G)) = 3$ and for any $k geq 2$, $I(G)^k$ has a linear resolution. More
In this article, we study the regularity of integral closure of powers of edge ideals. We obtain a lower bound for the regularity of integral closure of powers of edge ideals in terms of induced matching number of graphs. We prove that the regularity
Let $A = K[X_1,ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n colon J^infty)$ be the $n^{th}$ symbolic power of $I$ wrt $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, s