In 2008, Marino and his collaborator, suggested, based on mirror symmetry+Chern-Simons theory, that type A open string amplitudes in toric Calabi-Yau's, could be computed by the same recursion obeyed by large N expansion coefficients in matrix integrals. They had checked it in many cases with low genus, and it was proved to all genus only for the simplest 3D toric CY, namely $C^3$. par We recently proved this conjecture for all genus, for all toric CY. On the A-side, the localization formula rewrites the string amplitudes in terms of graphs. On the other side (B-side) of the conjecture, the topological recursion can also be encoded diagrammatically in terms of a sum of graphs. However, the graphs of the B-side have different weights on their edges and vertices, compared to the A-side. Also, graphs of the B-side have no (0,1) vertices=tadpole, and no (0,2) vertices. par Standard combinatorics manipulation of graphs, allow to introduce (0,1) and (0,2) vertices by changing the weights. One thus has to prove that those resummed weights are the same as those of the A-side. Using special geometry (Seiberg-Witten like relations), one shows that the weights of both side obey the same differential equation, and using tropical geometry one shows that at large Kahler parameters, the weights coincide. This proves the conjecture.