Abstract
A series of counting, sequence and layer matrices are considered precursors of classifiers capable of providing the partitions of the vertices of graphs. Classifiers are given to provide different degrees of distinctiveness for the vertices of the graphs. Any partition can be represented with colors. Following this fundamental idea, it was proposed to color the graphs according to the partitions of the graph vertices. Two alternative cases were identified: when the order of the sets in the partition is relevant (the sets are distinguished by their positions) and when the order of the sets in the partition is not relevant (the sets are not distinguished by their positions). The two isomers of C28 fullerenes were colored to test the ability of classifiers to generate different partitions and colorings, thereby providing a useful visual tool for scientists working on the functionalization of various highly symmetrical chemical structures.
Highlights
One set of works is especially related to the current study, since layer matrices were involved in the analysis of fullerenes: In [72], vertices were partitioned into classes of equivalence and ordered according to their centrality indexes, computed on layer matrices of vertex properties
The sequence and layer matrices involved here were previously reviewed in [9], and the coloring of vertices based on counting matrices was previously reported in [12]
Sequence and layer matrices are introduced accompanied with an example
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. One set of works is especially related to the current study, since layer matrices were involved in the analysis of fullerenes: In [72], vertices were partitioned into classes of equivalence and ordered according to their centrality indexes, computed on layer matrices of vertex properties. The use of the counting, sequence and layer matrices and some proposed modifications and extensions to generate different partitions on graphs are given and illustrated. These partitions were used for for getting visual representations of them. The sequence and layer matrices involved here were previously reviewed in [9], and the coloring of vertices based on counting matrices was previously reported in [12]
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