I am interested in the following: Given N vertices, how many different(up to graph-isomorphism) hypergraphs with every hyperedge covering exactly 3 different vertices, every vertex gets covered at least once and the hypergraph is connected exist?
For N <= 2: 0 because not enough vertices
N=3: 1 just {1,2,3} with numbers representing vertices and the set of numbers representing the hyperedge
N=4: 3 this {{1,2,3}, {2,3,4}} that {{1,2,3}, {2,3,4}, {1,3,4}} and {{1,2,4}, {2,3,4}, {1,2,3}, {1,3,4}}
N=5: more than 8
This question is quite similar to the following-only that I am a bit more restrictive. Number of different hypergraphs realizable on N vertices
I hope the question is clear.
Thanks!