  A one-by-one exhaustion is a combinatorial/geometric condition which excludes
eigenvalues from the spectra of Laplace and Schr\"odinger operators on graphs.
Isoperimetric inequalities in graphs with a cocompact automorphism group
provide an upper bound on the von Neumann dimension of the space of
eigenfunctions. Any finitely generated indicable amenable group has a Cayley
graph without eigenvalues.
