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Kappa Shape Indices

The Kappa Shape Indices

The kappa shape indices represent aspects of overall molecular shape [8]. The fundamental assumption that kappa indices are based on is that molecular shape is a function of the pattern in the network of skeletal bonds.  A scale for each order of index is established by referring to two extreme shapes.  For the first order index,1k, one extreme is given by the unbranched skeleton and the other by the graph with all atoms connected, called the complete molecular graph. The index is based on a count of bonds in the skeleton.  The number of bonds in the unbranched skeleton is called 1pmin; for the complete graph, 1pmax is used. For the molecule under consideration, 1-p-i is used as the count of skeletal bonds.

The first order kappa shape index is given as

The Kappa Shape Indices

The kappa shape indices represent aspects of overall molecular shape [8]. The fundamental assumption that kappa indices are based on is that molecular shape is a function of the pattern in the network of skeletal bonds.  A scale for each order of index is established by referring to two extreme shapes.  For the first order index,1k, one extreme is given by the unbranched skeleton and the other by the graph with all atoms connected, called the complete molecular graph. The index is based on a count of bonds in the skeleton.  The number of bonds in the unbranched skeleton is called 1pmin; for the complete graph, 1pmax is used. For the molecule under consideration, 1-p-i is used as the count of skeletal bonds.

The first order kappa shape index is given as:

The path counts in the two extreme structures can be simply related to atom count, N.  A factor is added to modify the expression to account for atom size, a, where:

Then the first order kappa alpha shape index is given as:

The information encoded in the three kappa alpha shape indices refers to attributes of shape. The first order index encodes the degree of cyclicity. The second order kappa index encodes the spatial density of atoms in the molecule. The third order kappa index encodes degree of centrality of branching. In practice only the kappa alpha indices are used as structure descriptors.

(See references 5 and 8 on our Methods and Descriptors Reference Page for additional information)

Examples of shapes and their corresponding k  indices:

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