Generalized Connectivity Indices

Figure 16. The plot of the first moment against the generalized connectivity index for different conformers of the chain of length five.

FIGURE 6.4 Graph on 20 vertices for which Table 6.11 has listed all paths of lengths two and their contrihutions to the second-order generalized connectivity index. 

Construction of the generalized connectivity index x(HD) for 3,4-dimethylheptane is illustrated in Table 9.2 at left. After making a list of bonds (the first column of... 

Molecular Connectivity Indexes and Graph Theory. Perhaps the chief obstacle to developing a general theory for quantification of physical properties is not so much in the understanding of the underlying physical laws, but rather the inabiUty to solve the requisite equations. The plethora of assumptions and simplifications in the statistical mechanics and group contribution sections of this article provide examples of this. Computational procedures are simplified when the number of parameters used to describe the saUent features of a problem is reduced. Because many properties of molecules correlate well with stmctures, parameters have been developed which grossly quantify molecular stmctural characteristics. These parameters, or coimectivity indexes, are usually based on the numbers and orientations of atoms and bonds in the molecule. 

Most chemicals used in the procedure will appear in the index. Thus, there will generally be entries for all starting materials, reagents, intermediates, important by-products, and final products. Most products shown in the Tables in the discus.sion sections of this volume are included unless the compounds are quite similar in which case a general descriptive name was entered. Chemicals generally nut indexed included coiimion solvents, standard inorganic acids and bases, reactants shown in the Tables, and compounds cited in the discussion section in connection with other methods of preparation. 

Kier and Hall (1976) and Hall et al. (1975) have pioneered the use of the connectivity index as a descriptor of molecular structure. It is an expression of the sum of the degrees of connectedness of each atom in a molecule. Indices can be calculated to various degrees or orders, thus encoding increasing information about the structure. Although the index has been used with success in a number of applications, it is not entirely clear on theoretical grounds why this is so. It appears that the index generally expresses molar volume or area. 

TTie extended edge connectivity indices were defined as a generalization of the edge connectivity index in analogy to the -> Kier-Hall connectivity indices ... 

Kupchik, E.J. (1989). General Treatment of Heteroatoms with the Randic Molecular Connectivity Index. Quant.Struct.-Act.Relat.,8, 98-103. 

Two molecular descriptors proposed to generalize the Randic connectivity index, defined as [Evans, Lynch et al, 1978]... 

Generalized topological indices are calculated by using common formulas of topological indices, where the exponent, if any, is allowed to differ from the standard value (e.g., —1/2 in the Randic connectivity index). Examples of these are the variable Zagreb indices, —> generalized connectivity indices, and —> generalized Wiener indices. 

The second atomic index [Figure 2.2(b)] is the valence connectivity index 8V, incorporating information on details of the electronic configuration of each non-hydrogen atom. Its value for the lowest oxidation states of the elements will generally be assigned by Equation 2.1 [2], where Zv is the number of valence electrons of an atom, NH is the number of hydrogen atoms bonded to it, and Z is its atomic number (i.e., Z equals Zv plus the number of inner shell electrons). 

The molecular connectivity indexes represent molecular structure in a manner analogous to the count of carbon atoms, but in a much more general way. That is, chi indexes are weighted counts of structure features with the same mathematical qualities as counts, but with much more structure information. 

The general name for the structural description method utilizing the adjacency relationships of molecular skeletons was selected to be molecular connectivity. The number assigned to a skeleton atom describing its adjacency relationship is called the simple connectivity value (or simple delta value) of the atom. In the development the S values were used for the first-order subgraph (or bond) between atoms i and j. The index for the entire molecule, in this case, the molecular connectivity index of the first order, is designated by the Greek letter chi, is computed as in equation 2,... 

Another useful notion is the so-called generalized bond index Kab referred to the given atoms A and B. According to [36], in case of CIS excited states, Kab, more exactly, Kab[C1S], turns out to be connected with CT numbers Ia b and some additional quantities. Before giving expressions applicable also to RPA and related models, we briefiy clarify the meaning of the generalized bond index. Even within the elementary MO theory, such as the tt-electron model, the corresponding quantity, that is /sT,uy[MO], is not the same as the usual bond order Instead, the squared bond order makes its appearance, namely, by definition the orbital index... 

Zhou and N. Trinajstic, On general sum-connectivity index, J. Math. Chem. 47 (2010b) 210-218. 

Quarter Century Symposium to celebrate 25 years of the publication of the article on the connectivity index. The symposium was attended by some 25 scientists for a two-day meeting offering an opportunity to participants to present some more recent results for the connectivity index [20]. Equally, as a significant development, one may mention various generalizations and extensions of the connectivity index, one of which is the variable connectivity index, which is discussed later in this chapter. 

E. J. Kupchik, General treatment of heteroatoms with the Randid molecular connectivity index, Quant. Struct.-Act. Relat. 8 (1989) 98-103. 

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