I have a question about hypergraphs that I hope some combinatorics/graph theory experts can answer. The motivation for this question is group-theoretic and comes from the study of a certain space of measures that comes equipped with a natural affine action of the group Out(F_n). I'll skip the detailed background here, but if someone is interested, please look-up my paper with Tatiana Nagnibeda arXiv:1105.5742

Let G be a finite simple graph. For every vertex v of G the link Lk(v) is the set of vertices adjacent to v. Now form a weighted hypergraph Lk(G) whose vertex set is the same as the vertex set of G and whose hyper-edges are exactly all sets Lk(v) as v varies over the vertex set of G. Every hyperedge E in Lk(G) comes with a positive integral "weight" w(E) which is the number of vertices of G such that Lk(v)=E.

Now suppose we are given a finite weighted hypergraph H with positive integral weights on its hyperedges. I'd like to know if there are known necessary and sufficient condition for H so that there exists a graph G such that H=Lk(G).

If anyone knows the answer or has some suggestions regarding where to look, I'd much appreciate it.

Thanks a lot,

Ilya Kapovich.