Extremum problems for eigenvalues of discrete Laplace operators

TL;DR

This paper investigates extremum problems for eigenvalues of discrete Laplace operators on polyhedral surfaces, identifying geometric shapes like equilateral triangles and squares as extremal cases for specific eigenvalue properties.

Contribution

It establishes geometric extremum results for eigenvalues of discrete Laplace operators on various polygons, highlighting the optimal shapes for maximal or minimal eigenvalue conditions.

Findings

Equilateral triangles maximize the first positive eigenvalue among all triangles.

Squares maximize the first positive eigenvalue among all cyclic quadrilaterals.

Regular n-gons minimize the sum and product of nontrivial eigenvalues among cyclic n-gons.

Abstract

The discrete Laplace operator on a triangulated polyhedral surface is related to geometric properties of the surface. This paper studies extremum problems for eigenvalues of the discrete Laplace operators. Among all triangles, an equilateral triangle has the maximal first positive eigenvalue. Among all cyclic quadrilateral, a square has the maximal first positive eigenvalue. Among all cyclic $n$-gons, a regular one has the minimal value of the sum of all nontrivial eigenvalues and the minimal value of the product of all nontrivial eigenvalues.