Path numbers

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

Randic M, Basak SC (1999) Optimal molecular descriptors based on weighted path numbers. J. Chem. Inf. Comput. Sci. 39 261-266. 

High tank level deviation DCS detects deviation DCS properly compensates Operator detects deviation Operator responds Consequences Path Numbers... 

Amic, D., Lucic, B., Nikolic, S., and Trinajstic, N. (2001) Predicting inhibition of microsomal p-hydroxylation of aniline by aliphatic alcohols a QSAR approach based on the weighted path numbers. Croat. Chem. Acta 74, 237-250. 

Fixed combinations of selected descriptors accounting for molecular properties of interest. The simplest combined descriptors are the differences and average values of basis descriptors such as -> connectivity indices or - path numbers, and the ratios of different descriptors defined with the aim of normalization to obtain, for example, size-independent indices. Moreover, optimal linear combinations of highly correlated descriptors are combined descriptors calculated so as to reduce the number of independent variables (e.g. - principal properties). 

A molecular weighted path number obtained by weighting graph edges with -> conventional bond order [Randic and Jurs, 1989] ... 

By summing all the elements of the molecular path code, the total path count P (also called total path number) is obtained ... 

One of the first - topological indices, introduced by H. Wiener in 1947 as the sum over all bonds of the product of the number of vertices on each side of the bond [Wiener, 1947c Wiener, 1947a Wiener, 1947b] and called at the beginning path number. Usually, the Wiener index is defined and calculated as the sum of all topological distances in the - H-depleted molecular graph [Hosoya, 1971]. 

Randic, M. (1997a). Linear Combinations of Path Numbers as Molecular descriptors. New J.Chem., 21,945-951. 

Randic, M. and Basak, S.C. (1999c). Optimal Molecular Descriptors Based on Weighted Path Numbers. J.Chem.InfComput.ScL, 39, 261-266. 

These are fixed combinations of selected descriptors accounting for molecular properties of interest. The simplest combined descriptors are the differences and average values of —> basis descriptors such as —> connectivity indices or —> path numbers, and the ratios of different... 

The molecular path count, also called path count, molecular path number or topological bond index with the symbol is the total number of paths oflength m in the graph and is denoted by... 

Wiener index (W) (= Wiener number, Wiener path number)... 

Lukovits, 1., Nikolic, S. and Trinajstic, N. (2002) On relationships between vertex-degrees, path-numbers and graph valence-shells in trees. Chem. Phys. Lett., 354, 417 22. 

Plavsic, D., Soskic, M., Landeka, I., Gutman, I. and Graovac, A. (1996a) On the relation between the path numbers Z, Z and the Hosoya Z index. 

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