On eigenfunctions and maximal cliques of Paley graphs of square order.

In this paper we present a family of maximal cliques of size q + 1 2 or q + 3 2 , accordingly as q ≡ 1 ( 4 ) or q ≡ 3 ( 4 ) , in Paley graphs of order q 2 , where q is an odd prime power. After that we use the new cliques to define a family of eigenfunctions corresponding to both non-principal eigenvalues and having support size q + 1 , which is the minimum possible value by the weight-distribution bound. Finally, we prove that the constructed eigenfunction comes from an equitable partition. [ABSTRACT FROM AUTHOR]