Elliptic operators on planar graphs: Unique continuation for   eigenfunctions and nonpositive curvature
  This paper is concerned with elliptic operators on plane tessellations. We show that such an operator does not admit a compactly supported eigenfunction, if the combinatorial curvature of the tessellation is nonpositive. Furthermore, we show that the only geometrically finite, repetitive plane tessellations with nonpositive curvature are the regular $(3,6), (4,4)$ and $(6,3)$ tilings. 
S. Klassert, D. Lenz, N. Peyerimhoff and P. Stollmann