NUMBER OF CLIQUES OF PALEY-TYPE GRAPHS OVER FINITE COMMUTATIVE LOCAL RINGS

In this work, given (R, m) a finite commutative local ring with identity and k E N with (k, |R|) = 1, we study the number of cliques of any size in the Cayley graph GR(k) = Cay(R, UR(k)) with UR(k) = {xk : x E R*}. Using the known fact that the graph GR(k) can be obtained by blowing -up the vertices of GFq (k) a number |m| of times, we reduce the study of the number of cliques in GR(k) over the local ring R to the computation of the number of cliques of GR/m(k) over the finite residue field R/m 'Fq. In this way, using known numbers of B -cliques of generalized Paley graphs (k = 2, 3, 4 and B = 3, 4), we obtain several explicit results for the number of B -cliques over finite commutative local rings with identity.