Answer from the Cytoscape development team...
Well, there are a number of ways you can use Cytoscape with TinkerPop, but I would think that the best way would be to use the REST interface to Cytoscape and inject your graph directly into it. You could save your graph as a Neo4j database and use the Neo4j app to pull it into Cytoscape or save your graph as a GraphML or some other common graph format and read it in that way. Neither Cytoscape or Gephi are UI frameworks, though -- both are standalone applications that are designed to import, visualize, analyze and manipulate very large graphs.
A good place to start is on the Cytoscape Automation topic in the Cytoscape manual: http://manual.cytoscape.org/en/stable/Programmatic_Access_to_Cytoscape_Features_Scripting.html