Hyper-Wiener WW(G) = ½·Σ[d(i,j) + d(i,j)²]. Randić (1993). Adds quadratic term to Wiener: pairs at large distance contribute disproportionately. WW = ½(W + W₂) where W₂ = Σd². More discriminating than W alone.
How to Use
Select graph:
- WW: Hyper-Wiener
- W+W₂: Components
- vs W: Compare
Discrimination Power
WW separates more graph pairs than W. The quadratic term amplifies large distances: if two graphs have same W but different distance distributions, WW will differ. Better for distinguishing similar structures.
Bounds
WW(K_n) = n(n-1)/2 (all d=1, d²=1). WW(P_n) = n(n-1)(n+1)(n+2)/24. WW ≥ W always. WW/W = (1+avg_d)/2 approximately.
Step-by-Step Instructions
- 1Select graph.
- 2Compute d(i,j) for all pairs.
- 3Sum d + d² per pair.
- 4Divide by 2.
- 5Compare with W.