  Let $G$ be a graph with edge ideal $I(G)$. We recall the notions of
$\min-match_{\{K_2, C_5\}}(G)$ and $\ind-match_{\{K_2, C_5\}}(G)$ from
\cite{sy}. We show that $${\rm reg}(I(G)^s)\leq 2s+\min-match_{\{K_2,
C_5\}}(G)-1,$$for all $s\geq 1$, which implies that$${\rm reg}(I(G)^s)\leq
2s+\min-match(G)-1.$$Moreover, we show that$${\rm reg}(I(G)^s)\geq
2s+\ind-match_{\{K_2, C_5\}}(G)-2,$$and if $\ind-match_{\{K_2, C_5\}}(G)$ is an
odd integer, then$${\rm reg}(I(G)^s)\geq 2s+\ind-match_{\{K_2,
C_5\}}(G)-1.$$Furthermore, it is shown that$${\rm reg}(I(G)^s)\leq
2s+\ord-match(G)-1,$$where $\ord-match(G)$ denotes the ordered matching number
of $G$. Finally, we construct infinitely many connected graphs which satisfy
the following strict inequalities:$$2s+\ind-match(G)-1 < {\rm reg}(I(G)^s)<
2s+{\rm cochord}(G)-1.$$This gives a positive answer to a question asked in
\cite{jns}.
