In the framework of regular Dirichlet forms we consider operators on infinite graphs. We study the connection of the existence of solutions with certain properties and the spectrum of the operators. 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.
