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All graph data structure can be serialized using the .NET serialization mechanism. You can also use the SerializeToBinary extension methods.
SerializeToBinary
var g = ...; // some graph using(var stream = File.Open("graph.bin")) g.SerializeToBinary(stream);
AdjacencyGraph<TVertex,TEdge> g; // some graph using(var stream = File.Open("graph.bin")) g = stream.DeserializeFromBinary();