Generalized Wiener number

Based on different powers of the topological distance, generalized expanded Wiener numbers IV", also called Tratch-Stankevitch-Zefirov-type indices, were proposed as [Klein and Gutman, 1999] ... 

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 ... 

Tratch-Stankevitch-Zefirov-type indices generalized expanded Wiener numbers -> expanded distance matrices... 

Polanski, J. and Bonchev, D. (1986). The Wiener Number of Graphs. I. General Theory and Changes Due to Graph Operations, MATCH (Comm.Math.Comp.Chem.),21,133-186. Polanski, J. and Bonchev, D. (1987). The Minimum Distance Number of Trees. MATCH (Comm. Math.Comp.Chem.), 21, 314-344. 

The (i, j)th elements of this matrix are given by the product of the number of vertices on each side of the path i, This definition is a straightforward generalization of the rule for construction of the Wiener number. The Wiener matrix for Gi is illustrated in Table 7. The sum of all entries above the main diagonal gives a hyper-Wiener number WW, which for Gi is 99. One can partition WW in contributions arising from atoms... 

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.)... 

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]. 

Apart from the use of the carbon number index, the first use of gri h invariants for the correlation of the measured properties of molecules with their structural features was made in 1947. In that yeax, Wiener [121,122] introduced two parameters designed for this purpose. The first of these was termed the path number and was defined as the "sum of the distances between any two carbon atoms in the molecule, in terms of carbon-carbon bonds. A simple algorithm was given for the calculation of this number and it was shown that its value for normal alkanes assumes the form - n). The second parameter was called the polarity number and was defined as "the number of pairs of carbcm atthree carbon-carbon bonds it took the general value n-3 f< normal alkanes. Wiener proposed that the variation of any physical property for an isomeric structure as compared to a normal alkane would be ven by the linear expression ... 

A more general process known as least-squares filtering or Wiener filtering can be used when noise is present, provided the statistical properties of the noise are known. In this approach, g is deblurred by convolving it with a filter m, chosen to minimize the expected squared difference between / and m g. It can be shown that the Fourier transform M of m is of the form (1///)[1/(1 - - j], where S is related to the spectral density of the noise note that in the absence of noise this reduces to the inverse filter M = /H. A. number of other restoration criteria lead to similar filter designs. 

Equations (1) and (2) can be useful in a number of practical cases (%,97). However, in most general cases, composite dielectrics are chaotic or statistical mixtures of several compoitcnts. Then the true value of permittivity of a statistic composite should lie between the values determined by Eq. (2) for n 1 and n —1. which is formulated by the Wiener inequalities [96),... 

The general PDM-based modular model evolved from the original Wiener-Bose modular model. It was first adapted to studies of neural systems that generate spikes (action potentials), whereby a threshold trigger is placed at the output ot the general modular model that obviates the use ot high-order kernels and yields a parsimonious complete model (Marmarelis 8c Orme 1993). The importance ot this development is found in that, ever since the Volterra-Wiener approach was applied to the study of spike-output neural systems, it had been assumed that a large number of kernels would be necessary to produce a... 

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