Resistance distance Ω(i,j): treat graph as electrical network, each edge = 1Ω resistor. Ω(i,j) = effective resistance between i and j. Klein-Randić (1993). Ω ≤ d(i,j) always. Ω uses ALL paths, not just shortest. More robust than geodesic distance.
How to Use
Select graph:
- Ω: Resistance dist.
- vs d: Geodesic
- Circuit: Network
Electrical Analogy
Each edge = 1Ω resistor. Apply voltage between i and j. Ω(i,j) = V/I = effective resistance. Parallel paths reduce resistance: more paths between i,j → lower Ω(i,j). Trees: Ω = d (no parallel paths).
Computation
Ω(i,j) = L⁺(i,i) + L⁺(j,j) - 2L⁺(i,j) where L⁺ = Moore-Penrose pseudoinverse of Laplacian. Can also use: Ω(i,j) = det(L[{i,j}])/det(L[{j}]). Elegant matrix formulas.
Step-by-Step Instructions
- 1Select graph.
- 2Build Laplacian.
- 3Compute L⁺.
- 4Calculate Ω(i,j).
- 5Compare with d(i,j).