On a metric property of perfect colorings

TL;DR

This paper proves a new metric property of perfect graph colorings, relating adjacency matrix rows to parameter matrix rows, with implications for various classes of perfect colorings and examples invalidating certain parameter matrices.

Contribution

It introduces a novel inequality linking adjacency and parameter matrices in perfect colorings, expanding understanding of their structural properties.

Findings

The L1 distance between adjacency matrix rows exceeds that of parameter matrix rows.

Corollaries for perfect 2-colorings and distance-regular graphs are derived.

Examples show certain parameter matrices are invalid for infinite graphs.

Abstract

Given a perfect coloring of a graph, we prove that the $L_1$ distance between two rows of the adjacency matrix of the graph is not less than the $L_1$ distance between the corresponding rows of the parameter matrix of the coloring. With the help of an algebraic approach, we deduce corollaries of this result for perfect $2$-colorings, perfect colorings in distance-$l$ graphs and in distance-regular graphs. We also provide examples when the obtained property reject several putative parameter matrices of perfect colorings in infinite graphs.