On weight distributions of perfect colorings and completely regular codes

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.

The minimum volume of subspace trades

“…and note that the value ϕ(Z), Z ∈ S (j) , is included s j,i times in the right part. This gives the recursive formula for W i ϕ , which is equivalent to (6). The next lemma follows from direct calculations, as well as from the symmetry.…”

“…Lemma 4 (see, e.g., [6]). Let ϕ be an eigenfunction, with the eigenvalue θ, of a connected graph, and let S be a completely regular set of vertices of the graph with the intersection numbers s i,i−1 , s i,i , s i,i+1 , i = 0, 1, .…”

To the theory of q-ary Steiner and other-type trades

“…To read more about how to calculate the weight distribution of eigenfunctions and generalizations of eigenfunctions, see [17]. Using Lemma 1, it is easy to derive the following lower bound on the support of an eigenfunction.…”

Local distributions for eigenfunctions and perfect colorings of q-ary Hamming graph

“…In case q > 2 the question is investigated in [1] for the 1-error-correcting codes. In [3] a more general case of the direct product of graphs is studied; however, the formula is not extended for the classes of graphs.…”