On a metric property of perfect colorings

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.