Kirchhoff index Kf(G) = sum of effective resistances Ω(i,j) over all pairs. Kf = n·Σ 1/μᵢ (Laplacian eigenvalues). Lower Kf = more network redundancy. Klein-Randić (1993). Connects graph theory to electrical network theory.
How to Use
Select graph:
- Kf: Kirchhoff index
- Ω: Resistance
- μᵢ: Laplacian
Effective Resistance
Model graph as electrical network: each edge = 1Ω resistor. Ω(i,j) = effective resistance between i and j. Metric! Triangle inequality holds. Captures 'electrical distance' — accounts for parallel paths.
Bounds
K_n: Kf = n-1 (minimum for n vertices). P_n: Kf = n(n²-1)/6 (maximum for trees). Star: Kf = (n-1)². Cycle C_n: Kf = n(n²-1)/12. Complete bipartite: clean formula.
Step-by-Step Instructions
- 1Select graph.
- 2Find Laplacian eigenvalues.
- 3Sum n/μᵢ.
- 4Compare bounds.
- 5Interpret resistance.