Detour index ω(G) = Σ l(i,j) over all pairs, where l(i,j) = longest path between i and j. Amić et al. (1998). The 'anti-Wiener': Wiener uses shortest paths, detour uses longest. ω ≥ W always. ω = W iff graph is a tree (unique paths).
How to Use
Select graph:
- ω: Detour index
- Longest: Per pair
- ω vs W: Compare
ω vs Wiener
W = Σ shortest paths. ω = Σ longest paths. For trees: ω = W (unique paths). For cycles: ω > W. The gap ω-W measures cycle 'richness' — how many alternative routes exist.
Complexity
Finding longest paths is NP-hard in general! But for small molecular graphs (n<50), exact computation is feasible. Detour matrix computation: O(n²·2ⁿ) worst case, practical for small n.
Step-by-Step Instructions
- 1Select graph.
- 2Find longest paths.
- 3Sum all l(i,j).
- 4Compare with W.
- 5Measure spaciousness.