How can a finite graph be viewed as a discrete analogue of a Riemann surface?

Solution 1:

What they are saying is somewhat true: A connected graph can be regarded as a metric space (when equipped with the graph metric) and as such can be used for approximation of geometry of Riemann surfaces (say, hyperbolic ones) as well as higher dimensional Riemannian manifolds. When done properly, one can derive some conclusions about, say, spectral properties of manifolds from spectral properties of approximating graphs. Take a look for instance at this paper by Brooks and Makover for some constructions and references. One of the most famous interactions of graph theory and theory of algebraic curves (over $\bar Q$) is via Belyi's theorem and "dessins d'enfants".

Edit: Two more connections between graphs and Riemann surfaces: (1) Tropical algebraic curves are graphs (equipped with some extra structure), (2) ribbon graphs can be used to describe Riemann surfaces equipped with certain holomorphic quadratic differentials. This is especially useful when analyzing topology of compactified moduli spaces of Riemann surfaces, see the linked n-Lab article for references.