  We present some applications of ideas from partial differential equations and
differential geometry to the study of difference equations on infinite graphs.
All operators that we consider are examples of "elliptic operators" as defined
by Y. Colin de Verdiere. For such operators, we discuss analogs of inequalities
of Cheeger and Harnack and of the maximum principle (in both elliptic and
parabolic versions), and apply them to study spectral theory, the ground state
and the heat semigroup associated to these operators.
