Resistance distance

In graph theory, the resistance distance between two vertices of a simple connected graph, G, is equal to the resistance between two equivalent points on an electrical network, constructed so as to correspond to G, with each edge being replaced by a 1 ohm resistance. It is a metric on graphs.

On a graph G, the resistance distance Ω_i,j between two vertices v_i and v_j is

where Γ is the Moore–Penrose inverse of the Laplacian matrix of G.

If i = j then

For an undirected graph

For any N-vertex simple connected graph G = (V, E) and arbitrary N×N matrix M:

From this generalized sum rule a number of relationships can be derived depending on the choice of M. Two of note are;

where the ${\displaystyle \lambda _{k}}$ are the non-zero eigenvalues of the Laplacian matrix. This unordered sum Σ_i<jΩ_i,j is called the Kirchhoff index of the graph.

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