Maximal Algebraic Connectivity for Paths with Fixed Effective Resistance

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.