Vertex connectivity index

Amic, D., Beslo, D., Lucic, B., NikoUc, S. and Trinajstic, N. (1998a). The Vertex-Connectivity Index Revisited. J.Chem.Inf.Comput.ScL, 38,819-822. 

Kezele, N., Klasinc, L., von Knop, J., Ivanis, S. and Nikolic, S. (2002) Computing the variable vertex-connectivity index. Croat. Chem. Acta, 75,651-661. 

Pogliani and his coworkers (Garcia-Domenech et al, 2008), besides illustrating the uses of the vertex-connectivity index and variable vertex-connectivity index, also... 

Randic introduced in 1991 the variable vertex-connectivity index (Randic, 1991a) using the concept of the augmented vertex-adjacency matrix (Randic, 1991a, 1991b Randic et al., 2001). Variants of adjacency matrices, called the augmented vertex-adjacency matrices and denoted by A, are the vertex-adjacency matrices that possess nonzero values on the main diagonal ... 

The summation of elements in the upper (or lower) matrix-triangle gives the vertex-connectivity index of G, (Gutman, 2002 Hu et al 2003). 

In 1991 Randic introduced the variable vertex connectivity index [31] as an alternative approach to Kier and Hall s index for characterization of heterosystems in QSPR-QSAR studies. The main difference between both molecular descriptors is that the former index uses optimized vertex weights (degf) and the latter uses fixed vertex weights (deg,) ... 

Hence, the variable connectivity index is a function of several variables and can assume numerical values only after particular values for x,y, and z are selected. In contrast, if we set x = 0,y = 0, and z = 0, we obtain the rigid index x for tryptophan. The variable vertex connectivity index remained overlooked... 

Koc represents the degree of sorption of polycyclic aromatic hydrocarbon in soil. Using three molecular parameters the length of molecule (L), the vertex connectivity index (Xy) and the edge connectivity index (Xe) and support vector regression with s = 0.05, following expression is obtained for the prediction of Kqc ... 

The Randic connectivity index, X, is also called the connectivity index or branching index, and is defined by Eq. (18) [7], where b runs over the bonds i-j of the molecule, and and dj are the vertex degrees of the atoms incident with the considered bond. 

A further extension of this approach was done by Kier and Hall8> so as to provide different values of the connectivity index for molecules depicted by one and the same graph, but differing by the chemical nature of atoms as well as by the presence of single, double or triple bonds. The valency of the atom i (vertex degree), Vj, is replaced by the atom connectivity ... 

The relation between this definition and the mathematical expression of and IIP values (Equation 5.13 and Equation 5.14) can be easily seen. The simple represents the vertex valence (a number of skeletal neighbors for each vertex). It can be presented as both = = k - h, and = - h, after the substitution of the number of valence electrons k with the number of electrons assigned to sigma orbitals . It is evident from Equation 5.15 that the greater the number of skeletal neighbors, the larger the value and the lower the connectivity index. Recently, new arguments were evaluated in support of the thesis that the molecular connectivity indices represent molecular accessibility areas and volumes (Estrada, 2002). 

The formerly proposed and the most important of this series of topological indices is the Baiaban distance connectivity index J (also called distance connectivity index or average distance sum connectivity). It is one of the most discriminating - molecular descriptors and its values do not increase substantially with molecule size or number of rings it is defined in terms of sums over each ith row of the - distance matrix D, i.e. the vertex distance degree o [Baiaban, 1982 Baiaban, 1983a]. It is defined as ... 

For example, the order local connectivity index for the ith vertex is defined as ... 

The first proposed electronegativity-based connectivity index was the DSI index [Diudea and Silaghi-Dumitrescu, 1989] based on the -> Sanderson group electronegativity ESG used as a local vertex invariant. It is defined as ... 

Among the -+ graph-invariants, several substituent descriptors have been defined such as, for example, - Kier steric descriptor, - steric vertex topological index and - fragment molecular connectivity indices. 

The first term % coincides with the - Randic connectivity index x the remaining values show monotonically smaller values. In acyclic graphs, the even powers of matrices necessarily make zero contributions as it is not possible to reach an adjacent vertex by an even number of steps. 

Calculation of the Balaban distance connectivity index] and Jt index for 2-methylpentane. D is the topological distance matrix distance sums and the vertex degrees. B equals 5 and C is zero. 

Relative chirality indices and chirality correction factors forthe two enantiomers shown below, together with valence vertex degrees and first-order valence connectivity index of the functional groups. 

Connectivity indices are among the most popular —> topological indices and are calculated from the —> vertex degree 5 of the atoms in the —> H-depleted molecular graph. The Randic connectivity index was the first connectivity index proposed [Randic, 1975b, 2008 Li and Gutman, 2006] it is also called connectivity index or branching index, and is defined as... 

Table C6 Values of the first-order Kier-Hall connectivity index for some substituent groups attached to a Carbon atom with a valence vertex degree equal to 3.

This index was designed as an extension of the Randic connectivity index to take into account the relative size of heteroatoms in a H-depleted molecular graph. It is based on the Madan vertex degree derived from the chemical adjacency matrix [Goel and Madan, 1995] ... 

Other molecular descriptors based on atom eccentricity values combined with other —> local vertex invariants are the eccentric connectivity index, the —> eccentric distance sum, the connective eccentricity index, the eccentric adjacency topochemical indices, the superadjacency index, and the —> eccentricity-hased Madan indices. 

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