How to Use
Select graph:
- W(G): Wiener index
- Average: Mean distance
- Compare: Different topologies
Chemical Applications
Wiener showed W correlates with boiling points, density, and surface tension of alkanes. Higher W → more 'stretched out' molecule → higher boiling point. This launched the field of chemical graph theory (topological indices).
Trees
For trees: W = Σ_{edges e} n₁(e)·n₂(e), where n₁,n₂ are the sizes of the two subtrees when edge e is removed. This gives an O(n) algorithm. Stars minimize W among trees; paths maximize it.
Step-by-Step Instructions
- 1Select graph.
- 2Compute W(G).
- 3Find average distance.
- 4Compare topologies.
- 5Check extremes.
Wiener Index Calculator — Frequently Asked Questions
Why is Wiener index useful in chemistry?+
Molecular structure affects physical properties. The Wiener index captures how 'compact' vs 'elongated' a molecule is. Linear alkanes (high W) have higher boiling points than branched isomers (low W). This was one of the first quantitative structure-property relationships.
Which graphs minimize/maximize W?+
Among n-vertex trees: star K_{1,n-1} minimizes W=(n-1)², path P_n maximizes W=n(n²-1)/6. Among all n-vertex connected graphs: K_n minimizes W=C(n,2). The path is always the worst case.
What's the average distance?+
d̄ = W(G)/C(n,2) = 2W/(n(n-1)). For random graphs: d̄ ≈ log n/log d̄. For small-world networks: d̄ is small despite large n. Social networks typically have d̄ ≈ 4-6.
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