Regularity of Powers of edge ideal of very well-covered graphs

Abstract

Let $k\ge 3$ be an integer and $G$ be a very well-covered graph with $\mathrm{odd}-\mathrm{girth}(G)\ge 2k+1$. Assume that $I(G)$ is the edge ideal of $G$. We show that for every integer $s$ with $1\le s\le k-2$, we have $\mathrm{reg}(I(G{)}^s)=2s+\nu (G)-1$, where $\nu (G)$ is the induced matching number of $G$.

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