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"].
