  We consider weighted graphs with an infinite set of vertices. We show that
boundedness of all functions of finite energy can be seen as a notion of
`relative compactness' for such graphs and study sufficient and necessary
conditions for this property in terms of various metrics. We then equip graphs
satisfying this property with a finite measure and investigate the associated
Laplacian and its semigroup. In this context, our results include the trace
class property for the semigroup, uniqueness and existence of solutions to the
Dirichlet problem with boundary arising from the natural compactification, an
explicit description of the domain of the Dirichlet Laplacian, convergence of
the heat semigroup for large times as well as stochastic incompleteness and
transience of the corresponding random walk in continuous time.
