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

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