On eigenfunctions and maximal cliques of Paley graphs of square order

In this paper we present a family of maximal cliques of size q+12 or q+32, accordingly as q≡1(4) or q≡3(4), in Paley graphs of order q2, 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.