Molecular connectivity indices types

The concept of the molecular connectivity index (originally called branching index) was introduced by Randic [266]. The information used in the calculation of molecular connectivity indices is the number and type of atoms and bonds as well as the numbers of total and valence electrons [176,178,181,267,268]. These data are readily available for all compounds, synthetic or hypothetical, from their structural formulas. All molecular connectivity indices are calculated only for the non-hydrogen part of the molecule [269-271]. Each non-hydrogen atom is described by its atomic 6 value, which is equal to the number of adjacent nonhydrogen atoms. For example, the first-order Oy) molecular connectivity index is calculated from the atomic S values using Eq. (38) ... 

For molecular connectivity indices with orders higher than 2, it is also necessary to specify the subclass of index. There are four subclasses of higher order indices path, cluster, path/cluster, and chain. These subclasses are defined by the type of structural subunits they are describing, a subunit over which the summation is to be taken when the respective indices are calculated. Naturally, the valence counterparts of all four subclasses of higher order indices can be easily defined by analogy, described above for the first-order valence molecular connectivity index. 

The main characteristic of cluster-type indices is that all bonds are connected to the common, central atom (star-type structure). The third-order cluster molecular connectivity index (3yc) is the first, simplest member of the cluster-type indices where three bonds are joined to the common central atom [102-104, 111-113,152-154,166,167,269]. The simplest chemical structure it refers to is the non-hydrogen part of ferf-butane. This index is then calculated using Eq. (43) ... 

Molecular descriptor proposed as the sum of atomic properties, accounting for valence electrons and extended connectivities in the H-depleted molecular graph using a Randic connectivity index-type formula [Lohninger, 1993] ... 

These topological indexes, based on the molecular connectivity approach, include three types the ""Xr molecular connectivity chi indexes that characterize the structural attributes of molecules, the ""k kappa indexes of molecular shape, and the topological equivalence state T values that individually characterize atoms and groups in the molecular skeleton and are used primarily to determine chemically equivalent atoms within a molecule. A further development of this approach has led to the electrotopological state atom indexes, which will not be discussed here but will be presented elsewhere. Molecular connectivity chi indexes are discussed in the first part of this paper along with illustrative applications. Then kappa shape indexes are discussed. The topological state index is discussed in the final section. 

Last but not least, we may mention considerable activity in some mathematical circles interested in mathematical properties of molecular descriptors. Mathematical Aspects of Randic-Type Molecular Structure Descriptors is the title of one of two books on the mathematical properties of the connectivity index X, in which selected papers on these topics were presented [21,22]. Clearly, the connectivity index represents one successfully solved problem in the search for useful mathematical molecular descriptors. But a question can be raised Is there anything unsolved relating to the topic of the connectivity index Is there something that can still be improved We will come up with some answers to these questions later. 

When the row sums as entries in construction of the connectivity-type indices are applied to the adjacency matrix of molecular graphs, one obtains the connectivity index % of Randid [10,11]. When they are applied to the graph distance matrix, one obtains Balaban s index J [12]. Both indices, x and J, have been very nseful in QSAR, which is an incentive to explore properties of additional structural indices of this type. At our disposal are several novel sparse matrices that have been recently introduced in chemical graph theory, which, in the view that their row sums cover a wider range of values, can be expected to produce novel highly discriminatory topological indices. 

A final useful index of sigma nonbonded interactions between lone pairs is the partial bond order p (Xm, Yn) which is evaluated over the MO s which result from the interaction of the lone pair group MO s with the sigma HOMO and vacant MO s of the coupling unit. This index is intimately connected with the type of analysis employed in this work. In our survey of a variety of problems of molecular structure we shall provide computational results pertinent to the analysis outlined, i e. all or some of the following indices will be provided ... 

Structure and data storage is shown on the right. A structure table contains the structures, their internal identifiers, and their external identifiers, if any. The structures are stored in a compact binary representation that includes the connection table, the coordinates, the ring information, and any stereochemical, valence, isomer, isotope, or bond information. Certain types of structure-specific information such as polymer or component designations are stored here, whereas other types of structure-specific information (atom- or bond-specific data, and more verbose text data) are stored in their own tables, referenced by the internal identifier, and the atom or bond numbers to which the data correspond. A formula table contains the molecular formula and various atom and atom-type indexes to enhance formula searching and sorting. 

The local curvature properties of the surface G(m) in each point r of the surface are given by the eigenv ues of the local Hessian matrix. Moreover, for a defined reference curvature b, the number p,(r, b) is defined as the number of local canonical curvatures (Hessian matrix eigenvalues) that are less than b. Usually b is chosen equal to zero and therefore the number p(r, 0) can take values 0,1, or 2 indicating that at the point r the molecular surface is locally concave, saddle-type, or convex, respectively. The three disjoint subsets Ao, Ai, and A2 are the collections of the surface points at which the molecular surface is locally concave, saddle-type, or convex, respectively the maximum connected components of these subsets Ao, Aj, and A2 are the surface domains denoted by Do,, Diand D2, where the index k refers to an ordering of these domains, usually according to decreasing surface size. 

The most popular complexity index was introduced by Bertz [Bertz, 1981 Bertz, 1983a Bertz, 1983b], tal g into account both the variety of kinds of bond connectivities and atom types of a - H-depleted molecular graph. 

The molecular shape indices (also called k, k, K-index) represent the overall molecular shape in three values based on counts of one-bond, two-bond, and three-bond fragments [58, 59]. Local topology (e.g., tetrahedral or planar coordination) as well as atom types are considered using parameter tables, including information such as valence radii and connectivity of the atoms. 

Atom-type topological indices are used to describe a molecule by information related to different atom types in the molecule. An atom-type index is usually derived from some properties of all the atoms of the same type and their structural environment. Atom-type Estate indices of Kier and Hall, perturbation connectivity indices, atom-type path counts, and atom-type autocorrelation descriptors are examples of these molecular descriptors. 

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