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Chemical graphs

XXXIV. Five new topological indices for the branching of tree-like graphs [1]

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Abstract

In order to find the centre of an acyclic connected graph (of a tree), vertices of degree one (endpoints) are removed stepwise. The numbers δi of vertices thus removed at each step form a digit sequenceS (pruning sequence) which reflects the branching of the tree. The sum of squares of digits in the sequenceS affords a new topologicalcentric index B = ∑i δ 2i for the branching of trees. Comparisons with other topological indices are presented evidencing thatB induces an ordering of isomeric trees distinct from those induced by all other indices devised so far, becauseB emphasizes equally branches of similar length.

It is shown that Rouvray's indexIis equivalent to Wiener's indexw, and that the Gordon-Scantlebury indexN 2 and Gutmanet al.'s indexM 1 belong to the same family, calledquadratic indices, and induce the same ordering.

Since all topological indices vary both with the branching and the number of vertices in the tree, four new indices are devised fromB andM 1 to account only (or mainly) for the branching, by normalization (imposing a lower bound equal to zero for chain-graphs, i.e.n-alkanes) or binormalization (same lower bound, and upper bound equal to one for star-graphs). Normalized and binormalized centric (C, C′) and quadratic indices (Q, Q′) are presented for the lower alkanes. From the five new topological indices, the centric indices (B, C, C′) are limited to trees, but the quadratic indices (Q, Q′) apply to any graph. Binormalized indices (C′,Q′) express the “topological shape” of the graph.

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  1. Organic Chemistry Department, The Polytechnic, Bucharest, Roumania

    Alexandru T. Balaban

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Balaban, A.T. Chemical graphs. Theoret. Chim. Acta 53, 355–375 (1979). https://doi.org/10.1007/BF00555695

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  • DOI: https://doi.org/10.1007/BF00555695

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