  We study the connection of the existence of solutions with certain properties
and the spectrum of operators in the framework of regular Dirichlet forms on
infinite graphs. In particular we prove a version of the Allegretto-Piepenbrink
theorem, which says that positive (super-)solutions to a generalized eigenvalue
equation exist exactly for energies not exceeding the infimum of the spectrum.
Moreover we show a version of Shnol's theorem, which says that existence of
solutions satisfying a growth condition with respect to a given boundary
measure implies that the corresponding energy is in the spectrum.
