Long paths in the distance graph over large subsets of vector spaces over finite fields

Let $E\subset {\mathbb{F}}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices corresponding to vectors $x,y\in E$ by an edge if $\mathrm{\mid }\mathrm{\mid }x-y\mathrm{\mid }\mathrm{\mid }={(x_1-y_1)}^2+\cdots +{(x_d-y_d)}^2=1$. We shall prove that if the size of $E$ is sufficiently large, then the distance graph of $E$ contains long non-overlapping paths and vertices of high degree.