Isoperimetric inequalities for eigenvalues of the Laplacian on cycles with fixed resistance metric

Abstract

For cycles with non-negative weights on its edges, we define its global resistance as the sum of the distances given by the effective resistance metric between adjacent vertices. We prove the following result: for the Laplace operator on the 3-cycle with global resistance equal to a given constant, the maximal value of the smallest positive eigenvalue and the minimal value of the largest eigenvalue, are both attained if and only if all the weights are equal to each other.