Finite graphs are a rich source of discrete analogues (I will be partially repeating the OP and some other answers here):
The Laplacian on a finite graph is a discrete analogue of the Laplacian on a Riemannian manifold. In particular, it is possible to formulate the heat equation, the wave equation, and the Schrödinger equation on a finite graph. There are actually two Laplacians, a vertex Laplacian and an edge Laplacian, which give a discrete analogue of Hodge theory.
The Ihara zeta function of a finite graph is a discrete analogue of the Selberg zeta function of a Riemannian manifold. A regular graph satisfies an analogue of the Riemann hypothesis if and only if it is a Ramanujan graph. There is also an analogue of the Selberg trace formula in this setting; Terras has written extensively about this kind of thing.
The Picard group (or critical group, or sandpile group) of a finite graph is a discrete analogue of the Picard group of an algebraic curve. More generally a lot of the theory of algebraic curves can be transported to this setting, e.g. the Riemann-Roch theorem.
(Finite graphs are also a rich source of other kinds of analogues; for example the Ihara zeta function is also analogous to the Dedekind zeta function of a number field, with coverings of graphs analogous to extensions of number fields and the Picard group analogous to the class group. There is even an analogue of the analytic class number formula in this setting although I have forgotten the reference.)
