  We consider the problem on finding the edge weights that maximize the
algebraic connectivity of a graph (smallest positive eigenvalue of the
Laplacian), subject to the condition that the total effective resistance is
kept constant. We propose the conjecture that for every graph the maximum is
attained for weights that are invariant under automorphisms. The solution to
the problem is given explicitly for the paths $P_3$ y $P_4$, where the
conjecture holds.
