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

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