Is there a method for associating the edges and vertices of an arbitrary directed graph with the irreducible representations of a compact Lie group and the intertwiners of the adjacent edge representations, respectively. In Physicists' terms is there a way to nterpret an arbitrary directed graph as a Penrose spin network
This is a very broad question and there is a lot I could say on the topic. I suggest as a starting point you read the book: Cvitanović, Predrag . Group theory. Birdtracks, Lie's, and exceptional groups. Princeton University Press, Princeton, NJ, 2008. 

