What's the standard graph representation file format or otherwise the most used one? I would say one of these three:
but it would be great if anyone could point out the advantages/weaknesses from each of them.
The answer ultimately lies with the size of your problem and what your are trying to achieve.
For example, none of those formats can handle billions of vertices whereas some dedicated large-scale graph analytics frameworks such as Spark or GraphLab-Create can.
GraphML and GEXF are roughly equivalent, both XML based. GraphML is standard and supported in a lot of graph librairies such as NetworkX, igraph, Boost Graph Library, Graph-tool, JGraphT, Gephi. GraphSon is not popular.
To draw your GraphML or GEXF graphs, you can use Gephi or Tulip (GEXF only).
I think JGF is a good candidate here too.
I evaluated a lot for a recent project and this came out on top. Specifically, I liked:
http://jsongraphformat.info/
https://github.com/jsongraph/json-graph-specification
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