  Let $k$ be a positive integer. We compute depth and Stanley depth of the
quotient ring of the edge ideal associated to the $k^{th}$ power of a path on
$n$ vertices. We show that both depth and Stanley depth have the same values
and can be given in terms of $k$ and $n$. For $n\geq 2k+1$, let $n\equiv
0,k+1,k+2,\dots, n-1(\mod(2k+1))$. Then we give values of depth and Stanley
depth of the quotient ring of the edge ideal associated to the $k^{th}$ power
of a cycle on $n$ vertices and tight bounds otherwise, in terms of $n$ and $k$.
We also compute lower bounds for the Stanley depth of the edge ideals
associated to the $k^{th}$ power of a path and a cycle and prove a conjecture
of Herzog presented in \cite{HS} for these ideals.
