A curvature notion for planar graphs stable under planar duality

Woess \cite{Woess98} introduced a curvature notion on the set of edges of a planar graph, called $\Psi$-curvature in our paper, which is stable under the planar duality. We study geometric and combinatorial properties for the class of infinite planar graphs with non-negative $\Psi$-curvature. By using the discharging method, we prove that for such an infinite graph the number of vertices (resp. faces) of degree $k,$ except $k=3,4$ or $6,$ is finite. As a main result, we prove that for an infinite planar graph with non-negative $\Psi$-curvature the sum of the number of vertices of degree at least $8$ and the number of faces of degree at least $8$ is at most one.