# HyperGraph & its Representation in Discrete Mathematics

A hypergraph can be described as a graph where, in place of connecting with two vertices/nodes, the hypergraph is connected with a subset of vertices/nodes. The edges can also be called hyperedges, which are used to contain arbitrary non-empty sets of vertices. These types of hyperedges are contained by the k hypergraph, which is connected with exactly k vertices. If there is a normal graph, it will be known as the 2 hypergraphs because in a normal graph, one edge is connected with the 2 vertices.

### Representation of Hypergraph

An undirected hypergraph H can be represented in a pair H = (V, E), where V is used to indicate a set of items, and these items are called vertices or nodes, and E is used to indicate a set of nonempty subsets of V, and these subsets are also called as edges or hyperedges.

Here E can be defined as a subset of P(X), where P(X) is used to indicate the Power set of X.

With the help of a closed curve, each hyperedge will be represented. The members of these hyperedges are contained by this closed curve so that it can create hypergraphs.

For example: Order & Size of Hypergraph

The order and size of the hypergraph can be determined with the help of following formula:

Order of Hypergraph = Size of vertex set, and

Size of Hypergraph = Size of the edges set

So in the above hypergraph, there is a total of 5 vertices named A, B, C, D, E, and the number of edges is 3 named: e1, e2, e3. As we know, Order of hypergraph = number of vertex and size of hypergraph = number of edges. So with the help of these values, the order and size of the above hypergraph are described as follows:

Hypergraph to Bi-Partite Graph

With the help of bipartite, we are able to always express a hypergraph, but expressing a hypergraph with the help of a bipartite graph is not always convenient. In this process, the hypergraphs are rarely utilized. In a bipartite graph, the set of a vertex is portioned into the two subsets named P and Q. Here each edge has a connection with a vertex in P to a vertex in Q. The vertices of H can be easily represented in the form of vertices in Q. The hyperedges of H can be easily represented in the form of vertices in P. If there is a case where s is a member of hyperedge t in H, then we will insert an edge (p, q). The above hypergraph is represented with the help of 2 ways. On the left side, the 5 edges are connected with the 3 hyperedges. The same 5 vertices are connected with the new vertices (three) on the right side, which is used to show the hyperedges by ordinary edges.

### Properties of Hypergraph

There are various types of properties contained by the hypergraph, which are described as follows:

Empty Hypergraph

The empty hypergraph does not contain any type of edges, but it can contain vertices.

Example: In the following diagram, we can see that there are no edges, but it contains five vertices named as: A, B, C, D, and E. d - Regular:

In this hypergraph, every vertex will contain a degree of d. This statement means that it is contained in precisely d hyperedges.

Example: In the below hypergraph, there are degree 2, which is contained by all the vertices (A, B, and C). That's why this hypergraph is 2 regular hypergraphs. 2- Colorable:

The vertices of 2-colorable are divided into the two classes named P and Q. With the help of these classes, each hyperedge with a cardinality of a minimum 2 contains the minimum 1 vertex from each class.

Non-Simple:

The non-simple hypergraph will contain loops, which can be defined as hyperedges with a single vertex or repeated edges, which can be defined as two or more than two edges containing the same set of vertices.

Example: In the following diagram, we can see that there are 2 loops named: e1 and e2. So this hypergraph is a non-simple hypergraph. Simple:

The simple hypergraph does not contain any loops or repeated edges.

K uniform:

In this, each hyperedge will be created with the help of exactly k vertices.

Example: In the below hypergraph, we can see that there are 4 hyperedges: e1, e2, e3, and e4, and each hyperedge has 2 vertices. So we can say that this hypergraph is a 2 uniform hypergraph. K partite:

In this hypergraph, each hyperedge is comprised into one vertex of each type, and the vertices in this hypergraph are divided into the k parts.

Example: In the below hypergraph, we can see that there are 3 parts in which vertices are divided, i.e., (A, D), (B, E), and (D, F). In this image, there is only 1 vertex contained by each hyperedge for each partition. ### Feedback   