Details

From the adjacency matrix (gmat in the code) of a graph with nodes, the matrix is constructed:

,

.

Denote the nodes between which the effective resistance is computed as and . The matrix is rearranged into a block structure: , where is a matrix carrying the resistance information for the nodes and , is a matrix, and is a matrix. By completion of squares, the effective resistance between nodes and is computed after integrating out the information of the other vertices to get the matrix . The inverse of an off-diagonal element in this matrix gives the effective resistance between a pair of nodes. You can think of the off-diagonal element as the cross term between nodes and , which is the inverse of the effective resistance after the other vertices are integrated out.

References

[1] Wikipedia. "Resistance Distance." (May 24, 2016) en.wikipedia.org/wiki/Resistance_distance.

[2] Wikipedia. "Completing the Squares." (May 24, 2016) en.wikipedia.org/wiki/Completing_the_square.

[3] F. Y. Wu, "Theory of Resistor Networks: The Two-Point Resistance," Journal of Physics A: Mathematical and General, 37(26), 2004, pp. 6653–6673. doi:10.1088/0305-4470/37/26/004.