On weight distributions of perfect structures

We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular code (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the percentage of the colors over the code. For some partial cases of completely regular codes we derive explicit formulas of weight distributions. Since any (other) completely regular code itself generates a perfect coloring, this gives universal formulas for calculating the weight distribution of any completely regular code from its parameters. 1 Perfect colorings and completely regular codes Let G = (V = {0, . . . , N−1}, E) be a graph; let f be a function (“coloring”) on V that possesses exactly k different values e0, . . . , ek−1 (“colors”). The function f is called a perfect coloring with parameter matrix S = (Sij)i,j=1, or S-perfect coloring, iff for any i, j from 0 to k − 1 any vertex of color ei has exactly Sij neighbors of color ej . (The corresponding partition of V into k parts is known as an equitable partition. In another terminology, see e.g. [3], the coloring f is called S-feasible and S is called a front divisor of G.) In what follows we assume that ei is the tuple with 1 in the ith position and 0s in the others (the length of the tuple may vary depending on the context; in the considered case it is k). Denote by A the adjacency matrix of G. Then it is easy to see [7, Lemma 9.3.1] that f is an S-perfect coloring if and only if