Extensions of the Wiener number
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Particularly for structure-property correlations there are many chemical graph-theoretic indices, one of which is Wiener's "path number". Because Wiener's original work focused on acyclic structures, one can imagine different ways of extending it to cycle-containing structures, several of which are noted here. Many of these different formulas in fact yield like numerical values for general connected graphs - that is, different formulas sometimes correspond to the same graph invariant. Indeed it is found that there are two "dominant" classes of formulas each corresponding to one of two distinct invariants. Extensions to sequences of invariants (with the Wiener index the first member) are more often found to give distinct sequences. Further a powerful vector-space theoretic view for characterizing and for comparing different sequences is described. This is illustrated for a collection of eight sequences in application to a set of molecular graph structures (corresponding to the octanes). Another type of extension is to generate a sequence or partially ordered set of graph invariants for which the Wiener index is the natural first member of this set. Certain such sets of invariants (corresponding to contributions from different types of subgraphs) are noted to be "complete" (or form a basis) in the sense that any invariant can be faithfully linearly expanded in terms of the members of the set.
JOURNAL OF CHEMICAL INFORMATION AND COMPUTER SCIENCES
author list (cited authors)
Zhu, H. Y., Klein, D. J., & Lukovits, I.
complete list of authors
Zhu, HY||Klein, DJ||Lukovits, I