Using Graph theory to show the resistance in an edge is ((2^n)-1)/(n*2^(n-1))

Question

Consider the hypercube {0,1}ⁿ, with 2ⁿ vertices, and with edges between vertices (a₁, . . . , a_n), (b₁, . . . , b_n) ∈ {0,1}ⁿ when they differ in exactly one coordinate. Show that the effective resistance across an edge is 2ⁿ−1/ n2ⁿ⁻¹

This question was on my graph theory final and I have NO idea how to approach it and solve it.. Any help would be greatly appreciated.

Answer 1

By Foster's theorem, the sum of all effective resistances should be equal to the number of nodes in the graph minus one, i.e., $\sum_e R_e =2^n-1$. By symmetry all edges have the same effective resistance. Since there are $n2^{n-1}$ edges, this gives the desired result