The Erd\"os-Falconer distance problem on the unit sphere in vector   spaces over finite fields
  art, Iosevich, Koh and Rudnev (2007) show, using Fourier analysis method, that the finite Erd\"os-Falconer distance conjecture holds for subsets of the unit sphere in $\mathbbm{F}_q^d$. In this note, we give a graph theoretic proof of this result. 
Le Anh Vinh