The Randic Connectivity Index

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 connectivity index, Xr similar to the M2 index was introduced by Randic5) for characterization of molecular branching  

This index was applied to correlations with gas chromatographic retention index, boiling points, standard enthalpies of formation in gas phase, heats of solution, refractive indices, theoretically calculated total surface area of alkanes. 

Here Vj, Vj. vh+1 are the degrees of the vertices in the path of length h. The index may also be obtained 7) from the h-th power of the adjacency matrix, Ah. 

The connectivity indices of order higher than three have not been used due to the expected small contribution to the molecular properties of the interactions between distant atoms. The calculation of 2yR for G1 is presented below  

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  

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Several physical properties of benzenoid hydrocarbons can be correlated with a function of the molecular connectivity of their equivalent Gutman trees. Namely, we use the Randic connectivity index [32], x, viz.,... 

Figure 9 shows a correlation between the heats of atomizations [37] of two families of unbranched hydrocarbons and the Randic connectivity index of their equinumerical (equivalent) caterpillars. 

J2) cj, where the sum runs over the occupied MOs for a single AO. The molecular mass is represented by M, and is the Randic connectivity index of order 3. 

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. 

Another topological descriptor specifically proposed for cis/trans isomerism is the Pogliani cis/trans connectivity index xcr> defined in terms of the - Randic connectivity index y as ... 

This interpretation places emphasis on the bimolecular encounter possibility among molecules, reflecting the collective influence of the bond accessibilities of each molecule with other molecules in its immediate environment. Therefore, the Randic connectivity index can be interpreted as the contribution of one molecule to the bimolecular interaction arising from the encounters of bonds of two identical molecules ... 

A variant of the Randic connectivity index was also proposed as ... 

These are modifications of the Randic connectivity index defined in such a way as to account for the presence of heteroatoms in the molecule [Kupchik, 1986 Kupchik, 1988 Kupchik, 1989] ... 

From the edge adjacency matrix, a graph-theoretical invariant analogous to the -> Randic connectivity index was derived by Estrada [Estrada, 1995a] it is called edge connectivity index, denoted by e, and defined as ... 

Molecular descriptors defined in analogy with the - Randic connectivity index and calculated on hydrogen-included molecular graphs where heavy vertices are weighted by valence group electronegativities [Diudea et al, 1996a]. 

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. 

The map connectivity matrices are another set of map matrices based on partitioning of the —> Randic connectivity index and the higher order —> connectivity indices into contributions arising from paths of length k [Randic and Basak, 2002]. They are defined as... 

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... 

The Randic connectivity index is closely related to the second Zagreb index M2 and was proposed as measure of —> molecular branching. 

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

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] ... 

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. 

Unlike matrix MM, which usually is unsymmetrical, the matrix H is symmetric and related to MM by a similarity transformation, so that H and MM have the same eigenvalues and interrelated eigenvectors. The matrix H is also related to the Laplacian matrix moreover, the half sum of the elements of H coincides with the Randic connectivity index ... 

The —> walk connectivity indices are molecular descriptors defined by analogy with the —> Randic connectivity index by using the atomic walk counts in place of the vertex degrees [Razinger, 1986]. 

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