ON A METRIC PROPERTY OF PERFECT COLORINGS

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 and perfect colorings in distance-l graphs and distance-regular graphs. We also provide examples of infinite graphs, where the obtained property rejects several putative parameter matrices of perfect colorings.