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"].