  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.
