g = Graph[{1 \[DirectedEdge] 3, 3 \[DirectedEdge] 4,
   4 \[DirectedEdge] 2, 2 \[DirectedEdge] 1, 2 \[DirectedEdge] 3},
  EdgeWeight -> {(1 \[DirectedEdge] 3) -> -2, (3 \[DirectedEdge] 4) ->
      2, (4 \[DirectedEdge] 2) -> -1, (2 \[DirectedEdge] 1) ->
     4, (2 \[DirectedEdge] 3) -> 3}]
vl = VertexList[g];
dm = GraphDistanceMatrix[g];
Grid[LexicographicSort[
  DeleteCases[
   Catenate[
    Table[{vl[[i]], vl[[j]], dm[[i, j]]}, {i, Length[vl]}, {j,
      Length[vl]}]], {x_, x_, _}]]]