CHEMICAL GRAPHS .34. 5 NEW TOPOLOGICAL INDEXES FOR THE BRANCHING OF TREE-LIKE GRAPHS

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 sequence S (pruning sequence) which reflects the branching of the tree. The sum of squares of digits in the sequence S affords a new topological centric index B = ∑i δi 2 for the branching of trees. Comparisons with other topological indices are presented evidencing that B induces an ordering of isomeric trees distinct from those induced by all other indices devised so far, because B emphasizes equally branches of similar length. It is shown that Rouvray's index Iis equivalent to Wiener's index w, and that the Gordon-Scantlebury index N2 and Gutman et al.'s index M1 belong to the same family, called quadratic 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 from B and M1 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. © 1979 Springer-Verlag."