Harary index H(G) = Σ 1/d(i,j) over all pairs. Reciprocal of distance: nearby pairs contribute most. Higher H = more compact graph. Named after Frank Harary. Correlates with molecular surface area and biological activity.
How to Use
Select graph:
- H: Harary index
- 1/d: Per pair
- Compact: Quality
H vs Wiener
Wiener W = Σ d(i,j): sum of distances. Harary H = Σ 1/d(i,j): sum of reciprocal distances. W penalizes distant pairs; H rewards nearby pairs. H and W give complementary structural information.
Bounds
K_n: H = n(n-1)/2 (maximum, all distances = 1). P_n: H = Σ_{k=1}^{n-1} (n-k)/k (harmonic-like sum). Star: H = (n-1) + (n-1)(n-2)/4.
Step-by-Step Instructions
- 1Select graph.
- 2Compute distances.
- 3Sum reciprocals.
- 4Compare compactness.
- 5Apply to molecules.