  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. There exists a finitely generated group G with
finite generating sets S and S' such that the adjacency operator of the Cayley
graph of (G,S) has no eigenvalue while the adjacency operator of the Cayley
graph of (G,S') has pure point spectrum.
