The Connectivity Index

In the mid-1970s, the connectivity index % (chi) was introduced in physical chemistry—if one views chemical graph theory and mathematical chemistry in a broad sense to be areas overlapping with physical chemistry. The connectivity 

Kier and Hall demonstrated very shortly after the connectivity index was proposed that this index, in combination with structurally related higher order connectivity indices, turned out to correlate well with many physical properties of molecules, in particular, alkanes, and a few molecular properties of molecules having heteroatoms, such as amines, alcohols, etc. In the next section, we outline the construction of the connectivity index. 

In the introductory part of the article On Characterization of Molecular Branching, [10] after the opening sentences, Some molecular properties depend upon molecular shape and vary regularly within a series of homologous compounds. The degree of branching of the molecular skeleton is the critical factor involved, the following can be found  

As is known, inequalities can have numerous solutions, so one should look for a simple solution, if there is such. One way to find solutions for a set of inequalities is to test a selection of simple expressions to see if they satisfy the inequalities. In our case, for example, one could consider (m, n) to be m = 1/m and n = 1/n, which is incidentally the choice of (m, ri) values used in the construction of one of the Zagreb indices [11-15]. However, this simple choice is not a solution as one can verify, because it does not satisfy all the seven inequalities. After testing a few similar expressions, it was found that by taking m = 1/Vm and n = 1/Vn all the seven inequalities of Table 6.1 are satisfied. This is the way the new bond additive molecular descriptor, the connectivity index x = 2H/. m n, was bom. The numerical values for the seven inequalities using the weights 1/V(m n) are shown in Table 6.1. In Table 6.2, we show additional solutions of the nine inequalities that offer an alternative form of the connectivity index x, which was more recently found but has not yet been tested. 

We should make two brief comments (i) One should keep in mind that m = 1/Vm and n = 1/Vn is one of the possible solutions of the set of considered inequalities, and in addition to those of Table 6.2, there may be additional solutions of interest in different applications, (ii) If some expressions, such as the mentioned Zagreb indices, do not satisfy the inequalities of Table 6.1 (based on 

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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. 

The values 1/V(dj dj) are for the atoms i and j, which make up this bond, and the connectivity index, x, is obtained as the sum of the bond connectivities. In molecules containing heteroatoms, the d values were considered to be equal to the difference between the number of valence electrons (E") and the number of hydrogen atoms (hi). Thus, for an alcoholic oxygen atom, d = 1, and d = 5. The valence connectivity-index, y can then be calculated the use of removes redundancies that can occur through the use of y alone. The calculation of connectivity indices and for the case of two isomeric heptanols is as follows. 

The 5 connectivity index (atom degree), that has a central role in computing the E-state, was used in the definihon of the Zagreb topological indices [11]. Randle modified the Zagreb index M2 to obtain the connectivity index % [12]. 

Kier and HaU extended the definition of the 5 connectivity index in order to incorporate heteroatoms and multiple bonds in the definition of the connectivity index % [13-15]. They noticed that the 5 connectivity (atom degree) may be expressed as ... 

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 ... 

This method is thus about equivalent in accuracy to the bond contribution method, but the connectivity index does contain information about molecular configuration or topology which is absent from the simple bond contribution method. It thus is inherently more likely to express differences between isomers. Its primary disadvantage is the need to deduce the indices, which can be difficult to the uninitiated. The indices lack physical meaning, which is worrisome to those who seek to understand fully the inherent nature and principles of the estimation method. 

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. 

Milan Randic (1975) proposed a branching index ( ), commonly referred to now as the connectivity index ... 

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). 

Randic, M., The connectivity index 25 years after, J. Mol. Graphics Modelling, 20, 19-35, 2001. 

The calculated retention is compared with the experimental value in Table 1. From a statistical point of view, the improvement when using the rotamer with better fit is insignificant (Fg g = 1.26 and 1.07) and the assumption could be considered as disputable. This can be explained by the small number of compounds that are able to rotate around the cr-bond in the total matrix. Independent of the reliability of the assumption, a beneficial conclusion can be drawn from Eqs. (l)-(3). They demonstrate that the retentions of the studied benzene derivatives are governed in the studied cases only by local descriptors and predominately by the local descriptors for the — O — atom in the hydroxyl group. The connectivity index used in many studies (e.g.. Ref. 5) showed, in the studied case, a correlation coefficient of only 0.285. 

Jinno and Kawasaki " correlated log k with the connectivity index y of 26 polycyclic aromatic hydrocarbons (PAHs) separated on phenyl (r=0.9926), ethyl (r=0.9905), octyl (r=0.9944), and octadecyl (r=0.9912)... 

Burda et al. separated 54 alkanes (C5-C11), with different degrees of branching, by RP-HPLC on an octadecy 1-silica Lichrosorb R18 column, by using a mixture of methanol and water (80 20) as mobile phase. The connectivity index x was employed to correlate the structures of investigated alkanes with their obtained retention parameter, log Jd. 

The difference matrix I-H is called the normalized Laplacian matrix Lnorm (also sometimes called just the Laplacian matrix) of G and there is much theory about it.230 The matrix Lnorm is clearly also related to the connectivity index ... 

Hunt et al. have used ab initio methods to study ion pairs in l-butyl-3-methylimidazolium (Bmim) ILs. The anions were Cl, BF4 , and NTf2". The authors established relationships between ion-pair association energy and a derived parameter called the connectivity index . Overall, the results suggest that Bmim-Cl forms a strongly connected and quite highly structured network, which leads to the rather high viscosity observed experimentally. In contrast, Bmim-NTf2 only forms a rather weak network, where the connectivity and the viscosity are much lower [106],... 

Another series of successfully applied topological descriptors is derived from graph theory using atom connectivity information of a molecule. An example is the connectivity index developed by Randic [21], In the simple form,... 

D denotes the degrees of the atoms i and j of a molecular graph, and the sum goes over all adjacent atoms. The exponent was originally chosen by Randic to be -1/2 however, it has been used to optimize the correlation between the descriptor and particular classes of organic compounds. The connectivity index was later extended by Kier and Hall to applications with connected subgraphs [22],... 

Kier and Hall Index is a special form of the connectivity index that allows optimizing the correlation between the descriptor and particular classes of organic compounds. 

Araujo, O. and De La Pena, J.A. (1998) Some bounds for the connectivity index of a chemical graph. 

Gutman, I., Araujo, O. and Morales, D.A. (2000b) Estimating the connectivity index of a saturated hydrocarbon. Indian J. Chem., 39, 381-385. 

Exponent-dependent properties of the connectivity index. Indian J. Chem., 41, 457 61. 

Hansen, P. and Melot, H. (2003) Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index. 

Peng, X.-L., Pang, K-T, Hu, Q.-N. and Liang, Y.-Z. (2004) Impersonality of the connectivity index and recomposition of topological indices according to different properties. Molecules, 9, 1089—1099. 

Table 4.3. Experimental change ACp of the heat capacity at the glass transition in J/(mole K), the geometrical parameter NBBrol used in the correlation, and the fitted values of ACp, for 89 polymers. The connectivity index 1%v, which is also used in the correlation, is listed in Table 2.2. The alternative set of values listed in Table 2.3 is used for the silicon-containing polymers. The 33 ACp(exp) values listed in parentheses are those for which there is a question mark in parentheses ( ) for the appropriate number of "mobile beads" in the tabulations [3,34],... 

Investigation of chromatographic retention is one of the most active areas for QSAR studies using connectivity indexes and other topological indexes. Many papers have been written for this important area of analytical chemistry. Sabljic demonstrated that the connectivity indexes are very useful in the QSAR analysis of chromatographic retention and lists several references. In particular, he refers to analysis of chlorinated alkanes and benzenes. For 13 chlorinated benzenes he showed that the first-order index gives excellent correlation (r = 0.997 and 0.985) for retention on SE-30 and on Carbowax 20M, respectively. These regressions were better than ones based on the use of the Wiener number or the Balaban ] index, even with a heteroatom modification. The isomer pairs 1,3- and 1,4-dichlorobenzene as well as the other isomers are not discriminated by the first-order index alone. 

Figire 11. Graphs showing degeneracy with respect to a selected invariant. Z = Hosoya Z topological index, x = the connectivity index, ID = identification number, and prime ID is the identification number based on prime number weights. 

The simple regression equation using the connectivity index % as the molecular descriptor for the smaller benzenoids is given in Table 26. To improve the regression, Radecki, Lamparczyk, and Kaliszan considered as an additional descriptor the ratio L/B of the length L and the width B of the rectangle within which the molecule... 

We would like to critically examine the interpretation of the regression equation (2) that it supports a notion that the Chromatographic retention index for benzenoids is shape dependent. As one can see from equation (1), most of the correlation between the structure and the Chromatographic retention index as the property is described by the connectivity index that is bond additive and size dependent. The conclusion therefore ought to be that the Chromatographic retention values are mostly bond additive and size dependent. Because of this pronounced bond-additivity, clearly we can understand why indices that are global are inferior in this particular application. 

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