Bell index B(G) = (1/n)Σ(d(v)-d̄)² where d̄ = 2m/n. Simply the variance of the degree sequence! Bell (1992). B = 0 ⟺ all vertices have the same degree ⟺ regular. B captures how 'spread out' vertex degrees are.
How to Use
Select graph:
- B: Bell/variance
- d̄: Mean degree
- B=0?: Regular
Statistical View
B is literally variance of {d(v)}. Standard deviation = √B. Coefficient of variation CV = √B/d̄. These statistical measures apply directly! A graph is 'more irregular' if CV is large.
Zagreb Relation
n·B = M₁ - 4m²/n. From: Σ(d-d̄)² = Σd² - n·d̄² = M₁ - 4m²/n. So B = M₁/n - (2m/n)². Elegant connection to first Zagreb!
Step-by-Step Instructions
- 1Select graph.
- 2Compute d̄ = 2m/n.
- 3For each v: (d(v)-d̄)².
- 4Sum and divide by n.
- 5Check if B=0.