Concise and elegant!
g=Graph[
{1\[DirectedEdge]2,1\[DirectedEdge]3,2\[DirectedEdge]3,
2\[DirectedEdge]4,3\[DirectedEdge]5,4\[DirectedEdge]3,
4\[DirectedEdge]5,4\[DirectedEdge]6,5\[DirectedEdge]6},
EdgeWeight->{1,12,9,3,5,3,13,15,4},
EdgeLabels->Rule@@@Transpose@{
{1\[DirectedEdge]2,1\[DirectedEdge]3,2\[DirectedEdge]3,
2\[DirectedEdge]4,3\[DirectedEdge]5,4\[DirectedEdge]3,
4\[DirectedEdge]5,4\[DirectedEdge]6,5\[DirectedEdge]6},
{1,12,9,3,5,3,13,15,4}},
VertexStyle->Yellow,VertexSize->0.5,
VertexShapeFunction->{{Yellow,Disk[#1,0.1],Black,Text[#2,#1]}&}]
route={{1,2,1},{1,3,12},{2,3,9},{2,4,3},{3,5,5},{4,3,4},{4,5,13},{5,6,4},{4,6,15}};
myroute=Select[Table[Select[route,#[[2]]==z&],{z,1,6}],#=!={}&]
Table[Set[v[temp],
SortBy[If[#[[1]]==1,{v[temp],#[[3]]},{v[temp],v[#[[1]]]+#[[3]]}]&/@
Select[Flatten[myroute,1],#[[2]]==temp&],Last][[1,2]]],{temp,2,6}]&/@Range[2]
