"Underlying graph of an Abstract Simplicial Complex hypergraph is the hypergraph's 2-section." It seems that graph associations (transformations), like this one between a graph and its simplex graph, are naturally described using hypergraphs. Graph powers is other case: resulting edges are encoding a subset of vertices contained in the path with size equal to the exponent of the power graph.
Very interesting thoughts! I agree that hypergraphs are useful in describing graph transformations. Another interesting thing to think about is how we can generalize graph transformations, and since you mentioned graph powers, I think it's that we can combine the graph k-th power and the simplex graph operation to give us a generalization of the simplex graph, where two cliques are adjacent when they differ by the addition/deletion of k vertices.
![What are Clique Complexes? [Hypergraph Episode 9]](http://i.ytimg.com/vi/Aoh4haV7Ekc/mqdefault.jpg)
![What are Clique Complexes? [Hypergraph Episode 9]](/img/tr.png)
"Underlying graph of an Abstract Simplicial Complex hypergraph is the hypergraph's 2-section." It seems that graph associations (transformations), like this one between a graph and its simplex graph, are naturally described using hypergraphs. Graph powers is other case: resulting edges are encoding a subset of vertices contained in the path with size equal to the exponent of the power graph.
Very interesting thoughts! I agree that hypergraphs are useful in describing graph transformations. Another interesting thing to think about is how we can generalize graph transformations, and since you mentioned graph powers, I think it's that we can combine the graph k-th power and the simplex graph operation to give us a generalization of the simplex graph, where two cliques are adjacent when they differ by the addition/deletion of k vertices.