Balaban Index

The Balaban index  is a graph index defined for a graph on  nodes,  edges, and  connected components by

where  is the circuit rank of the graph,  is the edge set, and  is the sum of all entries in the th row (or column) of the graph distance matrix.

Unless otherwise stated, hydrogen atoms are usually ignored in the computation of such indices as organic chemists usually do when they write a benzene ring as a hexagon (Devillers and Balaban 1999, p. 25).

Since disconnected graphs have an infinite element in each distance matrix row or column, summing gives infinity and taking the reciprocal gives 0 for each term, resulting in an overall value of 0.

Precomputed values for many graphs is implemented in the Wolfram Language as GraphData[g, "BalabanIndex"].

REFERENCES

Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N. "Resistance-Distance Matrix: A Computational Algorithm and Its Applications." Int. J. Quant. Chem. 90, 166-176, 2002.

Balaban, A. T. "Distance Connectivity Index." Chem. Phys. Lett. 89, 399-404, 1982.

Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 117-119, 1999.

Mercader, E.; Castro, E. A.; and Toropov, A. A. "Maximum Topological Distances Based Indices as Molecular Descriptors for QSPR. 4. Modeling the Enthalpy of Formation of Hydrocarbons from Elements." Int. J. Mol. Sci. 2, 121-132, 2001.

Randić, M. "Characterization of Molecular Branching." J. Am. Chem. Soc. 97, 6609-6615, 1975.