  We prove estimates relating exponential or sub-exponential volume growth of
weighted graphs to the bottom of the essential spectrum for general graph
Laplacians. The volume growth is computed with respect to a metric adapted to
the Laplacian, and use of these metrics produces better results than those
obtained from consideration of the graph metric only. Conditions for absence of
the essential spectrum are also discussed. These estimates are shown to be
optimal or near-optimal in certain settings and apply even if the Laplacian
under consideration is an unbounded operator.
