How to export and import Graphs with additional data?

I think it must be an oversight in the graph export code (it also happens when exporting to graph formats other than Graphlet). If you use explicit labels in Mathematica, it gets exported properly, just the implicit VertexLabels -> "Name" does not.

Note that since the export formats do not have an equivalent of VertexLabels -> "Name", you'd expect the export/import cycle to yield explicit VertexLabels.

So, I guess you should submit a bug report to Wolfram Research.

In the meantime, the work around of using explicit labels should not be too onerous; e.g.,

Graph[{1 -> 2, 2 -> 3, 3 -> 1}, VertexLabels -> Table[i -> i, {i, 3}],
  VertexStyle -> Purple, VertexShapeFunction -> "Diamond"]

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This can be automated in Export by using something like

ExportGraph[file_String, g_Graph, opts___] :=
 If[Flatten[{PropertyValue[g, VertexLabels]}] === {"Name"},
  Export[file, SetProperty[g,
         VertexLabels -> Table[i -> i, {i, VertexList[g]}]], opts],
  Export[file, g, opts]]

If you don't want to import the graph into some other application, but simply save it for later use, I've found that just saving the graph in .m format preserves everything just fine:

Export["graph.m", graph]

This is the only approach I've found that preserves property lists, labels, everything.