Let E⊂Fdq, 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∈E by an edge if ||x−y||=(x1−y1)2+⋯+(xd−yd)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.