On a metric property of perfect colorings
Abstract
Given a perfect coloring of a graph, we prove that the L1 distance between two rows of the adjacency matrix of the graph is not less than the L1 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.
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