Lanzhou index Lz(G) = Σ d(v)²·(n-1-d(v)). Named after Lanzhou University, China. Combines degree squared with 'deficiency' (n-1-d): how far from universal vertex. Lz(K_n) = 0 (all vertices are universal!). Innovative blend of Zagreb and complement perspectives.
How to Use
Select graph:
- Lz: Lanzhou index
- d²·def: Per vertex
- Lz=0?: Complete
Degree Deficiency
Deficiency = n-1-d(v): how many edges v is 'missing'. Complete: def=0. Star leaves: def=n-2. Lz combines large degree (d²) with large deficiency: vertices that are important BUT not fully connected.
Bounds
Lz(K_n) = 0 (min). Lz maximized by graphs with mixed high and low degree. For d-regular: Lz = n·d²·(n-1-d). Interesting: maximized at d ≈ 2(n-1)/3.
Step-by-Step Instructions
- 1Select graph.
- 2For each v: d²·(n-1-d).
- 3Sum contributions.
- 4Check if Lz=0.
- 5Analyze structure.