  For $E \subset {\Bbb F}_q^d$, $d \ge 2$, where ${\Bbb F}_q$ is the finite
field with $q$ elements, we consider the distance graph ${\mathcal
G}^{dist}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and
two vertices $x$, $y$ are connected by an edge if $||x-y|| \equiv
{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=t$. We prove that if $|E| \ge C_k
q^{\frac{d+2}{2}}$, then ${\mathcal G}^{dist}_t(E)$ contains a statistically
correct number of cycles of length $k$. We are also going to consider the
dot-product graph ${\mathcal G}^{prod}_t(E)$, $t \not=0$, where the vertices
are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if
$x \cdot y \equiv x_1y_1+\dots+x_dy_d=t$. We obtain similar results in this
case using more sophisticated methods necessitated by the fact that the
function $x \cdot y$ is not translation invariant. The exponent $\frac{d+2}{2}$
is improved for sufficiently long cycles.
