Cubic and quadruple Paley graphs with the nn-e.c. property

A graph G is n-existentially closed or n-e.c. if for any two disjoint subsets A and B of vertices of G   with |A∪B|=n|A∪B|=n, there is a vertex u∉A∪Bu∉A∪B that is adjacent to every vertex of A but not adjacent to any vertex of B. It is well-known that almost all graphs are n-e.c. However, few classes of n  -e.c. graphs have been constructed. A good construction is the Paley graphs which are defined as follows. Let q≡1(mod4) be a prime power. The vertices of Paley graphs are the elements of the finite field FqFq. Two vertices a and b are adjacent if and only if their difference is a quadratic residue. Previous results established that Paley graphs are n-e.c. for sufficiently large q. By using higher order residues on finite fields we can generate other classes of graphs which we called cubic and quadruple Paley graphs. We show that cubic Paley graphs are n  -e.c. whenever q⩾n224n-2q⩾n224n-2 and quadruple Paley graphs are n  -e.c. whenever q⩾9n262n-2q⩾9n262n-2. We also investigate a similar adjacency property for quadruple Paley digraphs.