Balaban J index: J = m/(μ+1) · Σ_{edges} 1/√(sᵢsⱼ) where sᵢ = row sum of distance matrix, μ = m-n+1 (cyclomatic number). Balaban (1982). Exceptional discrimination: rarely maps non-isomorphic graphs to same value. Better than most topological indices.
How to Use
Select graph:
- J: Balaban index
- sᵢ: Distance sums
- μ: Cyclomatic #
Discrimination Power
Among topological indices, J has the highest discrimination power: fewest pairs of non-isomorphic graphs with same J value. For trees up to 20 vertices: zero degeneracy! Approaches complete graph invariant quality.
Formula Details
sᵢ = Σⱼ d(i,j): distance sum. μ = m-n+1: cyclomatic (circuit) number. The factor m/(μ+1) normalizes for graph size and cyclicity. For trees: μ=0, so J = m·Σ 1/√(sᵢsⱼ).
Step-by-Step Instructions
- 1Select graph.
- 2Compute distance matrix.
- 3Find distance sums sᵢ.
- 4Compute J.
- 5Compare discrimination.