The hyper-Wiener index was recently introduced by Randic. The original definition given by Randié can be used for acyclic structures only. In this paper the definition of Randic was extended in two different fashions so as to be applicable for any connected structure. The formula provides an easy method to calculate the hyper-Wiener index for any graph.
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 graphsthat 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.
The effects of different types of boundaries on graphite fragments are considered as they influence the π-electrons. From a simple resonance theoretic argument there are proposed simple structural conditions governing the occurrence of "unpaired" π-electron density near the edges. Predictions based on these rules are made for a variety of edge structures. Further, the novel resonance theoretic argument and predictions are strengthened through more elaborate considerations of both the valence bond and molecular orbital theoretic nature, especially for translationally symmetric polymer strips with various types of edges.
Rules for molecular cyclicity based on the global indices resulting from reciprocal distances (Harary number, H) or from resistance distances (Kirchhoff number, Kf) were tested in comparison with those elaborated earlier by means of the Wiener index, W. The Harary number and the Wiener number were found to match molecular cyclicity in an almost identical manner. The Kirchhoff number also generally follows cyclicity trends described previously. H is slightly less degenerate than is W, but Kf has practically no degeneracy in the graphs investigated here. Being much more discriminating than the Wiener number (i.e., practically nondegenerate), Kf allowed the formulation of new rules for systems formed from linearly condensed ribbons of even-membered rings with different sizes as well as for branched ribbons. The topological cyclicity patterns are thus reformulated in an extended basis, proceeding from three different graph metrics.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.