The electrical resistance of a graph captures its commute and cover times
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Random walks on weighted graphs and applications to on-line algorithms
Journal of the ACM (JACM)
Graph Theory With Applications
Graph Theory With Applications
Note: Resistance distance and the normalized Laplacian spectrum
Discrete Applied Mathematics
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In this paper, using the intimate relations between random walks and electrical networks, we first prove the following effective resistance local sum rules: c"i@W"i"j+@?k@?@C(i)c"i"k(@W"i"k-@W"j"k)=2, where @W"i"j is the effective resistance between vertices i and j, c"i"k is the conductance of the edge, @C(i) is the neighbor set of i, and c"i=@?"k"@?"@C"("i")c"i"k. Then we show that from the above rules we can deduce many other local sum rules, including the well-known Foster's k-th formula. Finally, using the above local sum rules, for several kinds of electrical networks, we give the explicit expressions for the effective resistance between two arbitrary vertices.