Duality between Feynman graphs

[loop quantum gravity]

There's a definition of a dual graph to another graph, here:
https://en.wikipedia.org/wiki/Dual_graph

Now QFT at least the perturbative one is plagued with Feynman graphs, so obviously one can ask a trivial does the dual of a specific Feynman graph have a physical significance or physical meaning? or it doesn't add much more new physical information?

I once heard of the subject of Quantum graphs, I need a little bit more on this subject, I do know that it arises in Quantum Chaos.

 

[renormalize]

Now QFT at least the perturbative one is plagued with Feynman graphs, so obviously one can ask a trivial does the dual of a specific Feynman graph have a physical significance or physical meaning? or it doesn't add much more new physical information?

Google Scholar is your friend; give it a try!
Here's a 40 year old paper that addresses your question:
A duality property of planar Feynman diagrams
Abstract: It is found that the Fourier-transform of the amplitude of a planar Feynman diagram G can be written as the amplitude of the Feynman diagram G̃, where G̃ is the dual of G in the sense of graph theory, the propagators of G̃ being the Fourier-transformed of the ordinary ones.

 

[pre chronotopologo]

The interesting point may not be whether the dual graph contains “new” physical information, but what is preserved by the transformation G → G̃.
If the Fourier transform of the amplitude of a planar diagram G can be written as the amplitude of its graph-theoretic dual G̃, with Fourier-transformed propagators, then the question becomes more precise:
what mathematical structure is preserved under this dualization, and what are the precise conditions under which this correspondence ceases to hold?
That seems to me a more direct way of asking whether the dual has physical significance rather than being only an alternative representation.

---