Cycles of arbitrary length in distance graphs on ${\mathbb{F}}_q^d$

For $E\subset {\mathbb{F}}_q^d$, $d\ge 2$, where ${\mathbb{F}}_q$ is the finite field with $q$ elements, we consider the distance graph ${\mathcal G}^{dist}_t(E)$,$t \not=0$,wheretheverticesaretheelementsof$E$, and two vertices $x$, $y$ are connected by an edge if $||x-y|| \equiv {(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=t$\mathrm{.}Weprovethatif$|E| \ge C_k q^{\frac{d+2}{2}}$,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 ${\mathcal{G}}_t^{prod}(E)$, $t≠0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $x\cdot y\equiv x_1{y}_1+\cdots +x_d{y}_d=t$. 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.