Average Distance Sum Connectivity Index Balaban

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  

For simple graphs (saturated molecules) q is the number of edges. Averaged distance sums  

For multiple edges (unsaturated or aromatic systems) fractional distances are introduced in the distance matrix so that fractional distance sums result on summation if the bond order between vertices i, j is b, then 1/b is the entry in the row/ 

For all cycloalkanes having N = 2k carbon atoms, J = 2.0000 irrespective of k. It can be demonstrated that for linear alkanes J increases asymptotically with increasing N towards cycloalkanes with odd N tend asymptotically towards J = 2.0000 linear conjugated polyenes tend towards 4 /3, and linear conjugated 

The smallest two alkanes having degenerate J values have 22 vertices, where the total number of isomers (not counting stereoisomers) is 2,278,658. It will be seen, however, from Table 1 (see Sect. 4.7.) that the degeneracy for J is much lower than for any other single index. 

---