  In certain instances an electric network transforms in natural ways by the
addition or removal of an edge. This can have interesting consequences for
random walks, in light of the known relationships between electric resistance
and random walks. We exhibit several instances in which this can be used to
prove facts or simplify calculations. In particular, a new proof is given for
the formula for the expected return time of a random walk on a graph. We also
show how hitting times can be calculated in certain instances when a network
differs from a highly symmetric one by one edge.
