Quadrance polygons, association schemes and strongly regular graphs

Abstract

Quadrance between two points A_1 = [x_1,y_1] and A_2 = [x_2,y_2] is the number Q (A_1, A_2) := (x_2 - x_1)^2 + (y_2 - y_1)^2. In this paper, we present some interesting results arise from this notation. In Section 1, we will study geometry over finite fields under quadrance notations. The main purpose of Section 1 is to answer the question, for which a_1,...,a_n, we have a polygon A_1...A_n such that Q(A_i,A_{i+1})=a_i for i = 1,...,n. In Section 2, using tools developed in Section 1, we define a family of association schemes over finite field space F_q x F_q where q is a prime power. These schemes give rise to a graph V_q with vertices the points of F_q^2, and where (X,Y) is an edge of V_q if and only if Q(X,Y) is a nonzero square number in F_q. In Section 3, we will show that V_q is a strongly regular graph and propose a conjecture about the maximal clique of V_q.