  We prove that the edge ideals of line and cyclic graphs and their quotient
rings satisfy the Stanley conjecture. We compute the Stanley depth for the
quotient ring of the edge ideal associated to a cycle graph of length $n$,
given a precise formula for $n\equiv 0,2 \pmod{3}$ and tight bounds for
$n\equiv 1 \pmod{3}$. Also, we give bounds for the Stanley depth of a quotient
of two monomial ideals, in combinatorial terms.
