Total eccentricity ζ(G) = Σ ecc(v). Sum of maximum distances. ζ(K_n) = n (minimum: all vertices at distance 1). High ζ = spread out, peripheral graphs. Low ζ = compact. Simple but fundamental topological invariant.
How to Use
Select graph:
- ζ: Total ecc.
- ecc(v): Per vertex
- Compact: Low ζ
Bounds
ζ ≥ n (K_n achieves minimum). ζ ≤ n·(n-1) (achieved by... no connected graph, theoretical max). For trees: ζ maximized by path. ζ(P_n) ≈ 3n²/4 for large n.
Average Eccentricity
Average ecc = ζ/n. For K_n: avg=1. For C_n: avg=⌊n/2⌋. Measures 'typical' peripherality. Low average = compact network. High = extended.
Step-by-Step Instructions
- 1Select graph.
- 2BFS from each vertex.
- 3Record ecc(v) = max distance.
- 4Sum all ecc(v).
- 5Divide by n for average.