Hosoya polynomial counts finite sequences of distances in a graph G; more exactly, it counts the number of points/atoms lying at a given distance in G. The polynomial coefficients are calculable by means of layer/shell matrices. Shell matrix operator enables the transformation of any square matrix in the corresponding layer/shell matrix, thus generalizing the local property counting according to its distribution by the distances in G. This represents the “Hosoya-Diudea” generalized counting polynomial. We applied this theory to several hypothetical nanostructures with icosahedral symmetry.
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