  These are lecture notes for the Current Developments in Mathematics
conference at Harvard, November, 2011. We discuss topological, probabilistic
and combinatorial aspects of the Laplacian on a graph embedded on a surface.
The three main goals are to discuss: (1) for "circular" planar networks, the
characterization due to Colin de Verdi\`ere of Dirichlet-to-Neumann operator;
(2) The connections with the random spanning tree model; and (3) the
characteristic polynomial of the Laplacian on an annulus and torus.
