Current Developments in Mathematics
The Laplacian on planar graphs and graphs on surfaces
We discuss the Laplacian operator on graphs embedded on a surface, concentrating on simple surfaces (disk, annulus, torus). In particular we review and extend results on the reconstruction of the network from boundary measurements, connections with the spanning tree and grove model, and the characteristic polynomial of the Laplacian on an annulus and torus.
2010 Mathematics Subject Classification