Curvature, Geometry and Spectral Properties of Planar Graphs

We introduce a curvature function for planar graphs to study the connection of curvature and geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite, locally similar to a tessellation and admits no cut locus. For negative curvature we prove empty interior of minimal bigons and explicit bounds for the growth of distance balls and Cheeger’s constant. The latter yields bounds for the bottom of the spectrum for the discrete Laplace operator. More it leads to a characterization for triviality of essential spectrum by uniform decrease of curvature. Finally we show that non-positive curvature implies absence of finitely supported eigenfunctions for elliptic operators.