Inverse degree ID(G) = Σ 1/d(v). Simple yet powerful: leaves contribute 1 each, high-degree hubs contribute little. Also called zeroth-order Randić index. ID predicts: spanning tree count, chromatic number bounds, graph energy relationships.
How to Use
Select graph:
- ID: Inverse degree
- 1/d: Per vertex
- Spanning: Tree count
Properties
ID(K_n) = n/(n-1) ≈ 1 (hubs dominate). ID(star) = 1/(n-1) + (n-1) ≈ n (leaves dominate). High ID = many low-degree vertices. Low ID = many high-degree vertices. Simple but informative.
Conjectures
Fajtlowicz's AutoGraphiX: ID ≥ n/Δ. ID + diam ≥ n/δ. Many open conjectures involving ID and other graph parameters. Active research area.
Step-by-Step Instructions
- 1Select graph.
- 2For each vertex: 1/d(v).
- 3Sum contributions.
- 4Compare with n/Δ.
- 5Check conjectures.