The Wiener Index

For historical reasons the Wiener index, W, is introduced in this section. It was defined in 1947 and is still a starting point for the invention of new topological indices. 

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. 

Because of the symmetry of the distance matrix, the Wiener index can be expressed as Eq. (16). 

With Eq. (16) the Wiener index of compound 2 can be calculated from the distance matrix as shown in Eq, (17) 

Obviously, R = 2w, similarly to A = 2A for the total adjacency. Proceeding from this analogy one can call either w or R the total distance of the graph, giving a preference to the Wiener index since, due to the symmetry relative to the main matrix diagonal, the Rouvray index is redundant. Applications of w = R/2 are discussed in the following section. 

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Hence, they take account of only the structure constitution (topology). One of the first and most frequently used topological indices is the Wiener index. It has the form of Eq. (3), where D,y are all the routes from atom i to atom j. 

This is another index, which was independently developed by Polansky 46) for expressing some graph properties, including the relation with the Wiener index for important classes of molecular graphs, and by Bonchev, Balaban and Mekenyan 47)... 

Wiener Index The Wiener index, W, is defined as the sum of topological distances in G [15], considering either the upper right or lower left triangle of D ... 

One contra-intuitive feature of the Wiener index is that more distant atom pairs make a larger contribution to W than adjacent atom pairs (Randic and Zupan, 2001). A typical characteristic of W is that the central C-C bonds make a greater contribution than the peripheral bonds (Figure 5.5). For many bond additive physicochemical properties, such as boiling point, the opposite may be true, (i.e., the terminal bonds are considered more important to determine the magnitude of the property). 

Ivan Gutman, 50 Years of the Wiener Index, in. Serb. Chem. Soc., 62 (3), Serbian Chemical Society, Belgrade, 1997. 

The article reports investigations of the topological properties of benzenoid molecules which the author has performed in the last 20 years. Emphasis is given on recent developments and other scientists contributions to these researches. Topics covered in recent books and reviews are avoided. The article outlines spectral properties, some aspects of the study of Kekule and Clar structures, the Wiener index as well as a number of graphs derived from benzenoid systems (inner dual, excised internal structure, Clar graph, Gutman tree, coral and its dual). 

In addition to these, only a limited number of other topological indices of benzenoid molecules have been studied. With a few not too important exceptions, generally valid mathematical results were obtained only for one of them — namely for the Wiener index. Therefore the remaining part of this section is devoted to the Wiener index of benzenoid systems. (Further graph invariants worth mentioning in connection with benzenoids, especially unbranched catacondensed systems, are the Hosoya index [119-121], the Merrifield — Simmons index [122, 123], the modified Hosoya index [38] and the polynomials associated with them.)... 

The Wiener index has been extensively studied in the chemical literature (for reviews see [118, 124]) and therefore it is not surprising that it was also examined in the case of benzenoids. 

Recurrence relations have been established for the Wiener index of cataconden-sed benzenoid systems, both unbranched [125] and branched [126]. Based on these relations it could be shown that for unbranched catacondensed systems Uh with h hexagons [127],... 

Figure 23. The example of obtaining the values of the Wiener index for hypergraph and graph representations of the molecular structure with three-electron ligands hydrogens are suppressed in these representations...

S. Nikolic, N. Trinajstic, Z. Mihalic, The Wiener Index Developments and Applications, Croat. Chem. Acta, 68 (1995) 105-129. 

The investigation of chromatographic retention is one of the most active areas for QSRR studies using various topological indices. Many papers have been written for this important area of analytical chemistry.The first topological indices used for the prediction of retention parameters or lipophilic parameters were Randi s indices (molecular connectivity indices), the Wiener index, and the Balaban (7b) index. Selected applications of topological indices for the prediction of retention parameters of compounds separated by HPLC are covered later in this work. 

Adler et applied the Wiener index for calculating HPLC parameters for polycyclic hydrocarbons. Reten-... 

Amic et developed a simple relationship between experimental retention time by HPLC and the Wiener index (W), polarity number (p), and number of OH groups ( oh) in flavyliums ... 

Amic et al. also correlated the Schultz index MTI for anthocyanins. They suggest that the QSRR with MTI index should be identical to those with the Wiener index. 

It takes its name from the definition of the - Wiener index. 

Other encountered graphical bond order descriptors are the /V/ index, the W AV index, the WW AVW index, the J7J index, the CID7CID index, and the - Z7Z index derived, respectively, from the - Randic connectivity index, the -> Wiener index, the - hyper-Wiener index, the - Balaban distance connectivity index, the - Randic connectivity ID number, and the Hosoya Z index. 

IBC is divided by the Wiener index W rather than 2W in order to have AIBC values higher than one [Balaban et al, 1994b],... 

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