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

TL;DR

This paper establishes isoperimetric inequalities for Laplacian eigenvalues on weighted cycles with fixed resistance, showing optimal eigenvalues occur when all edge weights are equal.

Contribution

It proves that for a 3-cycle with fixed global resistance, the extremal eigenvalues are achieved with equal edge weights, extending isoperimetric principles to resistance metrics.

Findings

Maximal smallest positive eigenvalue occurs with equal weights

Minimal largest eigenvalue occurs with equal weights

Results apply specifically to 3-cycle graphs

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.