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.