Symmetric Hypergraphs
The rank of a hypergraph is the maximum cardinality of any of the edges in the hypergraph. If all edges have the same cardinality k, the hypergraph is said to be uniform or k-uniform, or is called a k-hypergraph. A graph is just a 2-uniform hypergraph.
The degree d(v) of a vertex v is the number of edges that contain it. H is k-regular if every vertex has degree k.
The dual of a uniform hypergraph is regular and vice-versa.
Two vertices x and y of H are called symmetric if there exists an automorphism such that . Two edges and are said to be symmetric if there exists an automorphism such that .
A hypergraph is said to be vertex-transitive (or vertex-symmetric) if all of its vertices are symmetric. Similarly, a hypergraph is edge-transitive if all edges are symmetric. If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply transitive.
Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity.
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