Use this answer by Yaroslav Bulatovthis answer by Yaroslav Bulatov.
test = LayeredGraphPlot[{1 -> 2, 1 -> 3, 2 -> 3, 1 -> 4,
2 -> 4, 1 -> 5}, VertexLabeling -> True, ImageSize -> 200]
getLabelCoordinateMap[gp1_] :=
Module[{points, labels, actualLabels, coordAliases, actualCoords},
points =
Cases[gp1, GraphicsComplex[points_, __] :> points, Infinity] //
First;
labels = Cases[gp1, Text[___], Infinity];
actualLabels = labels[[All, 1, 1]];
coordAliases = labels[[All, 2]];
actualCoords = points[[coordAliases]];
Thread[actualLabels -> actualCoords]];
coord = getLabelCoordinateMap[test]
Graph[{1 -> 2, 1 -> 3, 2 -> 3, 1 -> 4, 2 -> 4, 1 -> 5},
ImageSize -> 200, VertexLabels -> (# -> # & /@ Range@5),
VertexCoordinates -> coord, ImagePadding -> 10]
It gives the following: