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.

Cite: Krotov D.S.
On weight distributions of perfect colorings and completely regular codes
Designs, Codes and Cryptography. 2011. V.61. N3. P.315-329. DOI: 10.1007/s10623-010-9479-4 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Feb 1, 2010
Accepted:	Dec 14, 2010
Published online:	Dec 30, 2010

Identifiers:

Web of science:	WOS:000294297600006
Scopus:	2-s2.0-80052970028
Elibrary:	23957425
OpenAlex:	W3125362008

Citing:

DB	Citing
Web of science	24
Scopus	28
Elibrary	22
OpenAlex	32

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