  In this paper, we define (im, reg)-invariant extension of graphs and propose
a new approach for Nevo and Peeva's 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.
