Reading group PPT

1. Hypergraph
In mathematics, a hypergraph is a
generalization of a graph in which an edge
can join any number of vertices. Formally, a
hypergraph H is a pair H=(X,E) H = (X,E)
where X is a set of elements called nodes or
vertices, and E is a set of non-empty subsets
of X called hyperedges or edges. Therefore,
E is a subset of P(X) ,where P (X) is the
power set of X.

2. Hypergraph Definitions

3. Hypergraph Definitions
While graph edges are pairs of nodes, hyperedges are
arbitrary sets of nodes, and can therefore contain an
arbitrary number of nodes. However, it is often desirable
to study hypergraphs where all hyperedges have the same
cardinality; a k-uniform hypergraph is a hypergraph such
that all its hyperedges have size k. (In other words, one
such hypergraph is a collection of sets, each such set a
hyperedge connecting k nodes.) So a 2-uniform
hypergraph is a graph, a 3-uniform hypergraph is a
collection of unordered triples, and so on.

4. Hypergraph Variances
• A simple hypergraph is that all edges are
distinct. A simple hypergraph is a Hypergraph
H=(V;E) such that :𝑒𝑖 ⊆ 𝑒𝑖 if i=j. A simple
hypergraph has no repeated hypergraph.
• Hypergraph types: Directed Hypergraph,
Undirected Hypergraph

5. Undirected Hypergraph
• Let H=(V;E) be a hypergraph
• V={v1,v2,….vn} and E={e1,e2,…,em}
• With ∪ 𝑒𝑖 = V
• Then H has an 𝑛 × 𝑚 incidence matrix A=(𝑎𝑖𝑗)
where 𝑎𝑖𝑗=1 if vi ∈ 𝑒𝑗 otherwise 𝑎𝑖𝑗=0

6. Directed Hypergraph

7. Directed Hypergraph

8. Directed Hypergraph

9. Directed Hypergraph

10. Directed Hypergraph

11. Directed Hypergraph

12. Directed Hypergraph

13. Directed Hypergraph

14. Directed Hypergraph
• Hypergraph Property : Helly property;
Conformality property

15. Directed Hypergraph

16. Directed Hypergraph