Woess　[30]　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.　(C)　2021　Elsevier　Inc.　All　rights　reserved.