Your argument will give a bound involving the number of resistors in the network, which means the bound will grow with the number of resistors. However, I believe the bound should be independent of that.
Sure! I'm new to here. What I'm considering is a graph with banded adjacency matrix, which means that for entries close to the diagonal they are non zero and for entries far away from the diagonal they are 0. The entries in the adjacency matrix is just the conductance of that edge.
Suppose we have a connected electrical network with only resistors. We also assume that each vertex is connected to at least ##k## vertices, where ##k\geq2##. Suppose all resistors in the network has resistance between 1 and 2. Suppose we flow 1 unit of current from vertex ##a## to vertex ##b##...