In [[mathematics]], a [[hypergraph]] is a generalization of a [[graph (mathematics)|graph]], where edges can contain any finite number of vertices, and theThe '''line graph of a hypergraph''' is a generalization of the [[linegraph (mathematics)|graph]] of a graph. The precise definition is that the '''line graph''' of a hypergraph is the graph whose vertex set is the set of edgesthe hyperedges of the [[hypergraph]], with two edges adjacent when they have nonempty intersection. ThusIn other words, the line graph of a hypergraph is the same as the [[intersection graph]] of a family of finite sets. It is a generalization of the [[line graph]] of a graph.
However, the questions asked tend to be different. Questions about line graphs of hypergraphs are often generalizations of questions about line graphs of graphs. For instance, a hypergraph whose edges all have size ''k'' is called ''k''' ''-uniform'''. (A 2-uniform hypergraph is a graph.). In hypergraph theory, it is often natural to require that hypergraphs be ''k''-uniform. Every graph is the line graph of some hypergraph, but, given a fixed edge size ''k'', not every graph is a line graph of some ''k''-uniform hypergraph. A main problem is to characterize those that are, for each ''k'' ≥ 3.
A hypergraph is '''linear''' if each pair of hyperedges intersects in at most one vertex. Every graph is the line graph, not only of some hypergraph, but of some linear hypergraph {{harv|Berge|1989}}.
