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

Let $k\ge 3$ be an integer and $G$ be a very well-covered graph with ${\rm odd-girth}(G)\geq 2k+1$\mathrm{.}Assumethat$I(G)$istheedgeidealof$G$. We show that for every integer $s$ with $1\le s\le k-2$, we have ${\rm reg}(I(G)^s)=2s+\nu(G)-1$,where$\nu (G)$istheinducedmatchingnumberof$G$.