Distance degree

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 ... 

Criterion 2D for the vertices satisfying the first criterion, minimum -> vertex distance degree a... 

Similar to the previously defined connectivity indices but relative to the -> geometry matrix G, they are defined using the -> geometric distance degree in place of the topological vertex degree 6 ... 

The maximum/minimum path sum of the i th vertex, denoted by MmPVS, is a local vertex invariant defined as the sum of the lengths of the longest and shortest paths between vertex v, and any other vertex in the molecular graph. It is calculated as the sum of elements over the / th row and / th column in the A/D matrix, or, alternatively, as the sum of the - vertex distance degree o, calculated on the distance matrix D and the maximum path sum MPVS, of the / th vertex calculated on the detour matrix A ... 

The row sums of this matrix contain information on the molecular folding in fact, in highly folded structures, they tend to be relatively small as the interatomic distances are small while the topological distances increase as the size of the structure increases. Therefore, the average row sum is a molecular invariant called the average distance/ distance degree, i.e. 

From the frequencies of the row entries, the vertex distance code (or distance degree sequence of a vertex) is defined as the ordered sequence of the occurrences of the increasing distance values for the i th considered vertex,... 

The vertex distance degree (or distance number, distance index, distance rank, vei tex distance sum, distance of a vertex) is the row sum a, of the distance matrix D ... 

All these quantities are -> local vertex invariants. High values of the vertex distance sum o are observed for -> terminal vertices while low values for -> central vertices. Moreover, among the terminal vertices, the vertex distance degrees are small if the vertex is close to a branching site and larger if the terminal vertex is far away. 

Topological distance, distance degrees, eccentricities, topological radius and diameter, and frequencies are used to calculate several - topological information indices... 

The average row sum of the distance matrix is a molecular invariant called the average distance degree defined as ... 

The minimum value of the vertex distance degrees is another molecular invariant called the unipolarity ... 

Another molecular descriptor is the PRS index (or Product of Row Sums index), defined as the product of the vertex distance degrees a, ... 

The edge distance degree (or edge distance index, edge distance sum) is the row... 

The sum of the edge distance degrees, i.e. the sum of all matrix elements, is called total edge distance De and defined as ... 

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