hypergraph

hypergraph

A

hypergraph

ℋ

is an

ordered pair

(

V

,

ℰ

)

where

V

is a set of

vertices

and

ℰ

is a set of edges such that

ℰ

⊆

𝒫

⁢

(

V

)

. In other words, an

edge

is nothing more than a set of vertices. A hypergraph is the same thing as a

simple incidence structure

, but with terminology that focuses on the relationship with graphs.

Sometimes it is desirable to restrict this

definition

more. The empty hypergraph is not very interesting, so we usually accept that

V

≠

∅

.

Singleton

edges are allowed in general, but not the empty one. Most applications consider only finite hypergraphs, but occasionally it is also useful to allow

V

to be

infinite

.

Many of the definitions of graphs carry verbatim to hypergraphs.

ℋ

is said to be

k

-

uniform

if every edge

e

∈

ℰ

has

cardinality

k

, and is

uniform

if it is

k

-uniform for some

k

. An ordinary graph is merely a

2

-

uniform hypergraph.

The

degree

of a vertex

v

is the number of edges in

ℰ

that contain this vertex, often denoted

d

⁢

(

v

)

.

ℋ

is

k

-

regular

if every vertex has degree

k

, and is said to be

regular

if it is

k

-regular for some

k

.

Let

V

=

{

v

1

,

v

2

,

…

,

v

n

}

and

ℰ

=

{

e

1

,

e

2

,

…

,

e

m

}

. Associated to any hypergraph is the

n

×

m

incidence matrix

A

=

(

a

i

⁢

j

)

where

a

i

⁢

j

=

{

1

if

⁢

v

i

∈

e

j

0

otherwise

For example, let

ℋ

=

(

V

,

ℰ

)

, where

V

=

{

a

,

b

,

c

}

and

ℰ

=

{

{

a

}

,

{

a

,

b

}

,

{

a

,

c

}

,

{

a

,

b

,

c

}

}

. Defining

v

i

and

e

j

in the

obvious

manner (as they are listed in the sets), we have

A

=

(

1

1

1

1

0

1

0

1

0

0

1

1

)

Notice that the sum of the entries in any column is the cardinality of the corresponding edge. Similarly, the sum of the entries in a particular row is the degree of the corresponding vertex.

The transpose

A

t

of the incidence matrix also defines a hypergraph

ℋ

*

, the

dual

of

ℋ

, in an obvious manner. To be explicit, let

ℋ

*

=

(

V

*

,

ℰ

*

)

where

V

*

is an

m

-

element

set and

ℰ

*

is an

n

-element set of subsets of

V

*

. For

v

j

*

∈

V

*

and

e

i

*

∈

ℰ

*

,

v

j

*

∈

e

i

*

if and only if

a

i

⁢

j

=

1

.

Continuing from the example above, the dual

ℋ

*

of

ℋ

consists of

V

*

=

{

x

,

y

,

z

,

t

}

and

ℰ

*

=

{

{

x

,

y

,

z

,

t

}

,

{

y

,

t

}

,

{

z

,

t

}

}

, where

v

j

*

and

e

i

*

are defined in the obvious manner.

By the remark immediately after the definition of the incidence matrix of a hypergraph, it is easy to see that the dual of a uniform hypergraph is regular and vice-versa. It is not rare to see fruitful results emerge by considering the dual of a hypergraph.

Note

A finite hypergraph is also related to a metacategory, and can be shown to be a special case of a

supercategory

, and can be thus defined as a mathematical

interpretation

of

ETAS axioms

.

Title

hypergraph

Canonical

name

Hypergraph

Date of creation

2013-03-22 13:05:29

Last modified on

2013-03-22 13:05:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

24

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C65

Synonym

metacategory

Synonym

multi-graph

Synonym

colored graph

Related topic

SteinerSystem

Related topic

IncidenceStructures

Related topic

ToposAxioms

Related topic

TopicEntryOnTheAlgebraicFoundationsOfMathematics

Related topic

GraphTheory

Related topic

ETAS

Related topic

AxiomsOfMetacategoriesAndSupercategories

Defines

incidence matrix

Defines

uniform hypergraph

Defines

regular hypergraph

Generated on Fri Feb 9 07:55:20 2018 by

LaTeXML