On weight distributions of perfect colorings and completely regular codes

TL;DR

This paper investigates the weight distributions of perfect colorings and completely regular codes in graphs, providing formulas and methods for their computation, especially in Hamming graphs, advancing understanding of equitable partitions.

Contribution

It introduces a method to compute weight distributions of perfect colorings relative to completely regular sets, including explicit formulas for certain cases and a simple formula for Hamming graphs.

Findings

Derived explicit formulas for weight distributions in specific cases.

Provided a universal method to calculate weight distributions from parameters.

Proved a simple formula for the weight enumerator in Hamming graphs.

Abstract

A vertex coloring of a graph is called "perfect" if for any two colors $a$ and $b$, the number of the color-$b$ neighbors of a color-$a$ vertex $x$ does not depend on the choice of $x$, that is, depends only on $a$ and $b$ (the corresponding partition of the vertex set is known as "equitable"). A set of vertices is called "completely regular" if the coloring according to the distance from this set is perfect. By the "weight distribution" of some coloring with respect to some set we mean the information about the number of vertices of every color at every distance from the set. We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular set (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 color composition over the set. For some partial cases of completely regular sets, we derive explicit formulas of weight distributions. Since any (other) completely regular set itself generates a perfect coloring, this gives universal formulas for calculating the weight distribution of any completely regular set from its parameters. In the case of Hamming graphs, we prove a very simple formula for the weight enumerator of an arbitrary perfect coloring. Codewords: completely regular code; equitable partition; partition design; perfect coloring; perfect structure; regular partition; weight distribution; weight enumerator.