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Extremum problems for eigenvalues of discrete Laplace operators
The P1 discretization of the Laplace operator on a triangulated
polyhedral surface is related to geometric properties of the surface. This paper studies extremum problems for eigenvalues of the P1 discretization of the
Laplace operator. 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 positive eigenvalues and the minimal value
of the product of all positive eigenvalues.Keywords: extremum, discrete Laplace operator, spectra, cyclic polygo