Wiener hyper

Zhou, B. and Gutman, I. (2004b) Relations between Wiener, hyper-Wiener and Zagreb indices. Chem. Phys. Lett., 394, 93-95. 

For any square symmetric A x A matrix M representing a molecular graph with A vertices, the hyper-Wiener operator 3-Cy"W is defined as [Ivanciuc et al, 1997a] ... 

It takes its name from the definition of the hyper-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. 

For acyclic graphs, CJDp = CJAp = WW = Dp SZDp, where WW is the -> hyper-Wiener index. Dp the hyper-distance-path index, and SZDp the - hyper-Szeged index, while for cyclic graphs all these descriptors are different. 

Harary indices and -> hyper-Harary indices are defined for these matrices applying the Wiener operator. 

The hyper-detour index ww can be obtained applying the Wiener operator to the detour-path matrix, as ... 

For acyclic graphs, the hyper-detour index ww is equal to the index Dp obtained from the distance-path matrix Dp and to the WW obtained from the - Wiener matrix W. 

Applying the Wiener operator results in -> Harary indices and - hyper-Harary indices for these matrices and for the corresponding order -> sparse matrix, respectively. 

The hyper-distance-path index Dp can be obtained applying the - Wiener operator yv to the distance-path matrix ... 

For acyclic graphs the hyper-distance-path index Dp coincides with the - hyper-Wiener index WW derived from the - Wiener matrix and with the -> hyper-detour index derived from the detour-path matrix. Moreover, it was proposed as an extension of the hyper-Wiener index for any graph [Klein et al., 1995],... 

Note that n = 0 and n = 1 result in, respectively, the - hyper-Wiener index and the expanded Wiener number formally, for n -> -oo, the generalized expanded Wiener number should coincide with the well-known Wiener index. For cycle-containing graphs, the generalized expanded Wiener numbers were calculated as ... 

By generalization, Harary indices and hyper-Harary indices (or hyper-Harary numbers) are all molecular descriptors derived from the application of the Wiener operator to reciprocal topological matrices Harary indices are obtained from the 1 -order - sparse matrix, i.e. considering only the graph edges, while the hyper-Harary indices are from the whole matrices, i.e. considering the paths [Diudea, 1997c]. 

The Harary index and hyper-Harary index, defined only for acyclic graphs, are obtained from, respectively, the P -order sparse -> reciprocal Wiener matrix W ... 

For acyclic graphs, the equality between the - hyper-distance-path index Dp and the -> hyper-Wiener index WW is not true for the corresponding hyper-Harary indices, i.e. //dp / //ww-... 

In acyclic graphs, the Harary Szeged index //sz coincides with the Harary Wiener index Hvv ( sz = while the corresponding hyper-Harary indices are different (Hsz H v) in cyclic graphs all these indices differ [Diudea et al, 1997f]. 

A list of molecular identification numbers is given below. Other important ones are the Hosoya ID number, the -> hyper-Wiener index, the - restricted walk ID number, and the total topological state, defined elsewhere. 

Moreover, a number of topological indices have been expressed as functions of the Wiener index or as its extensions [Zhu et ai, 1996b] examples are - detour/Wiener index, - hyper-Wiener index, -> Harary index, - total information content on the distance magnitude, mean information content on the distance magnitude, and - mean information content on the distance degree magnitude. 

For acyclic graphs, the hyper-Wiener index is also obtained by applying the Wiener operator to the symmetric Cluj-distance matrix CJD and the -+ Wiener orthogonal operator to the Cluj-distance matrix CJDu-... 

The resistance distance hyper-Wiener index R has been proposed [Klein et al, 1995] based on the same general formula as that for the hyper-Wiener index, where the topological distance dij is replaced by the - resistance distance Q ... 

It should be noted that the hyper-distance-path index and the resistance distance hyper-Wiener index are equivalent to the hyper-Wiener index as defined by Randic for acyclic graphs, but they can also be applied to any cycle-containing connected graphs. 

Hyper-Harary indices are defined for this matrix and - Harary indices for the corresponding 1 order sparse matrix applying the Wiener operator. Moreover, the Wiener delta matrix was also proposed as W = W - W [Ivanciuc et al., 1997]. 

Examples of Wiener-type indices are -+ detour index, - Kirchhojf number or quasi-Wiener index, - Szeged index, -> all-path Wiener index, -> edge Wiener index, - hyper-Wiener index, -+ Cluj-distance index, - hyper-Cluj-distance index, -> Cluj-detour index, -> hyper-Cluj-detour index, -> Horary indices, -> hyper-Harary indices, -> detour/Wiener index, and - hyper-distance-path index. 

Diudea, M.V. and PSrv, B. (1995). Molecular Topology. 25. Hyper-Wiener Index of Dendrimers. J.Chem.lnf.Comput.ScL, 35,1015-1018. 

Diudea, M.V. (1996b). Wiener and Hyper-Wiener Numbers in a Single Matrix. J.Chem.lnf.Com-put.ScL, 36,833-836. 

Gutman, I., Linert, W., Lukovits, 1. and Dobrynin, A.A. (1997a). TVees with External Hyper-Wiener Index Mathematical Basis and Chemical Applications. J.Chem.Inf.Comput.Sci., 37, 349-354. 

Klein, D.J., Lukovits, I. and Gutman, I. (1995). On the Definition of the Hyper-Wiener Index for Cycle-Containing Structures. J.Chem.Inf.Comput.ScL, 35,50-52. 

Linert, W., Kleestorfer, K., Renz, F. and Lukovits, I. (1995a). Description of Cyclic and Branched-Acyclic Hydrocarbons by Variants of the Hyper-Wiener Index. J.Mol.Struct.(Theo-chem), 337,121-127. 

Linert, W, Renz, E, Kleestorfer, K. and Lukovits, I. (1995b). An Algorithm for the Computation of the Hyper-Wiener Index for the Characterization and Discrimination of Branched Acyclic Molecules. Computers Chem., 19,395-401. 

Linert, W. and Lukovits, I. (1997). Formulas for the Hyper-Wiener and Hyper-Detour Indices of Fosed Bicyclic Structures. MATCH (Comm.Math.Comp.Chem.), 35. 

Lukovits, I. and Linert, W. (1994b). A Novel Definition of the Hyper-Wiener Index for Cycles. J.Chem.lnfComput.Sci., 34, 899-902. 

Lukovits, I. (1995a). A Formula for the Hyper-Wiener Index. In QSAR and Molecular Modelling Cocepts, Computational Tools and Biological Applications (Sanz, F., Giraldo, J. and Man-aut, F., eds.), Prous Science, Barcelona (Spain), pp. 53-54. 

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