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 thi...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.2102.01958 |