Extremum problems for eigenvalues of discrete Laplace operators

► 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 quadrilaterals, a square has the maximal first positive eigenvalue. ► Among all cyclic n-gons, a regular one has the minimum of the sum and the product of nontrivial eigenvalues.