  Given a finite field, one can form a directed graph using the field elements
as vertices and connecting two vertices if their difference lies in a fixed
subgroup of the multiplicative group. If -1 is contained in this fixed
subgroup, then we obtain an undirected graph that is referred to as a
generalized Paley graph. In this paper we study generalized Paley graphs whose
clique and chromatic numbers coincide and link this theory to the study of the
synchronization property in 1-dimensional primitive affine permutation groups.
