  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.
