Upper bounds for eigenvalues of the discrete and continuous Laplace operators (Q1909929)

The authors study upper bounds for the eigenvalues of the Laplace operator on a compact Riemannian manifold and on a finite graph. They develop a universal approach for both continuous and discrete structures using the heat kernel. One of their results may be stated as follows. Let \(X\) and \(Y\) be disjoint subsets of the underlying manifold or graph \(M\). Let \(\mu\) denote the associated measure. Then \[ \lambda_1 \leq 4 \text{ dist}(X,Y)^{-2} (\text{ln}\{2\mu (M)/ \sqrt{\mu(X) \mu(Y)}\})^2. \] The logarithm comes from a Gaussian exponential term which enters the heat kernel upper bounds.