Distance Sum Index

This is another index, which was independently developed by Polansky 46) for expressing some graph properties, including the relation with the Wiener index for important classes of molecular graphs, and by Bonchev, Balaban and Mekenyan 47) 

The values of these distance sums are given below at each vertex in graphs Gj and G2  

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DN2Sy Triplet index from distance matrix, square of graph order ( of nonhydrogen atoms), and distance sum operation y — 1-5... 

Balaban applied 47a b> a Randic-type formula (7) to distance sums instead of vertices and created thereby the most discriminating (least degenerate) topological index proposed so far ... 

Balaban Index The Balaban index, J, has been introduced as the average-distance sum defined as the sum of the squares of the reciprocal nonzero distances in D (6,7] ... 

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 ... 

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 ... 

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... 

Balaban, A.T. and Filip, P.A. (1984). Computer Program for Topological Index J (Average Distance Sum Connectivity). MATCH (Comm.Math.Comp.Chem.), 16,163-190. 

The J index for multigraphs is calculated by the distance sums of the multigraph distance matrix D where the distances are obtained by weighting each edge with the reciprocal of its conventional bond order (—> relative topological distance) ... 

Calculation of the Balaban distance connectivity index] and Jt index for 2-methylpentane. D is the topological distance matrix distance sums and the vertex degrees. B equals 5 and C is zero. 

Other molecular descriptors based on atom eccentricity values combined with other —> local vertex invariants are the eccentric connectivity index, the —> eccentric distance sum, the connective eccentricity index, the eccentric adjacency topochemical indices, the superadjacency index, and the —> eccentricity-hased Madan indices. 

The distance-path matrix Dp for ethylbenzene. VS, indicates the matrix row sums. Dp is the hyper-distance-path index. 

The variable Balaban index, denoted as J, is calculated by analogy with the Balaban distance connectivity index from the row sums of the variable augmented distance matrix as [Randic and Pompe, 2001b]... 

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