Relation between regularity of powers of edge ideals and (im, reg)-invariant extension

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. Moreover, we consider new conjectures related to the regularity of powers of edge ideals of gap-free graphs.