association scheme

association scheme

Association schemes

were introduced by statisticians in the 1950’s to analyze designs of statistical experiments. Today, association schemes are useful not only in experimental designs, but in other areas of mathematics such as

combinatorics

(

coding theory

) and

group theory

(

permutation groups

).

There are several

equivalent

ways to define an association schemes. Three useful ones are illustrated here:

Let

Ω

be a non-empty set with

n

elements

, and

s

a positive integer.

Definition 1

. An

association scheme

𝒬

on

Ω

is a

partition

on

Ω

×

Ω

into sets

C

0

,

C

1

,

…

,

C

s

called

associate classes

, such that

•

each

C

i

is a

symmetric relation

on

Ω

, and

C

0

in particular is the

diagonal relation

,

•

for

i

,

j

,

k

∈

{

0

,

1

,

…

,

s

}

, there is an integer

p

i

⁢

j

k

such that, for any

(

a

,

b

)

∈

C

k

,

|

{

c

∈

Ω

∣

(

a

,

c

)

∈

C

i

⁢

and

⁢

(

c

,

b

)

∈

C

j

}

|

=

p

i

⁢

j

k

.

If we write

C

⁢

(

a

,

b

;

i

,

j

)

for the set

{

c

∈

Ω

∣

(

a

,

c

)

∈

C

i

⁢

and

⁢

(

c

,

b

)

∈

C

j

}

, then the second condition says that for any

(

a

,

b

)

∈

C

k

, the value

|

C

⁢

(

a

,

b

;

i

,

j

)

|

is a

constant

, depending only on

i

,

j

and

k

, and not on the particular elements of

C

k

. This implies that, for any

i

,

j

∈

{

0

,

1

,

…

,

n

}

, the

relation

C

i

∘

C

j

is a union of (some of) the

C

k

’s.

The definition above can be restated in graph theoretic terminology. First, think of

Ω

is a set of vertices, and two-element subsets of

Ω

are edges. The

complete graph

on

Ω

is just the set of all two-element subsets of

Ω

. We may color the edges of the graph. Say there are colors labeled

1

through

s

. For each color

i

, let

C

i

be the set of edges with color

i

. Then each

C

i

is just a symmetric relation on

Ω

, and that all the

C

i

’s, together with the diagonal relation, partition the set

Ω

×

Ω

. This is basically the first condition of the definition above. In this regard, we can redefine an association scheme graph theoretically, as follows:

Definition 2

. An

association scheme

𝒬

is a

surjective

coloring

on the

edge set

of a complete graph whose

vertex set

is

Ω

, by a set of

s

colors (numbered

1

through

s

), such that

for any

i

,

j

,

k

∈

{

1

,

…

,

s

}

, there is an integer

p

i

⁢

j

k

such that if

𝒬

⁢

(

a

,

b

)

=

k

(the edge

{

a

,

b

}

has color

k

), then

|

{

c

∈

Ω

∣

𝒬

⁢

(

a

,

c

)

=

i

⁢

and

⁢

𝒬

⁢

(

c

,

b

)

=

j

}

|

=

p

i

⁢

j

k

.

In words, the definition says that, for any color

k

, and any given edge

e

with color

k

, the number of

triangles

(a triangle in a graph is a cycle consisting of three edges) with

e

as an edge, and two other edges with colors

i

,

j

respectively, is

p

i

⁢

j

k

.

The first definition can also be viewed in terms of matrices, and

adjacency matrices

more specifically. Given a

finite set

Ω

, a

binary relation

R

on

Ω

naturally corresponds to matrix

A

called the adjacency matrix of

R

. Entry

(

i

,

j

)

is

1

if the

i

-th element and the

j

-th element are related by

R

, and

0

otherwise. If

R

is

reflexive

, then

A

has all

1

’s in its

diagonal

, and if

R

is

symmetric

, then so is

A

. Also, it is easy to see that the

composition

of two binary relations is the same as the

product

of their corresponding adjacency matrices. Then the comment in the paragraph after the first definition is the same as saying that the adjacency matrix of

C

i

∘

C

j

is a

linear combination

of the adjacency matrices of

C

0

,

C

1

,

…

,

C

s

. This gives us the third definition below:

Definition 3

. An

association scheme

is a finite set

𝒬

of

n

×

n

non-zero matrices

A

0

,

A

1

,

…

,

A

s

whose entries are

0

’s and

1

’s, such that

•

each

A

i

is a

symmetric matrix

, with

A

0

=

I

n

, the

identity matrix

,

•

A

0

+

A

1

+

⋯

+

A

s

=

J

n

, the matrix whose entries are all

1

’s, and

•

for any

i

,

j

∈

{

0

,

…

,

s

}

,

A

i

⁢

A

j

is a linear combination of

A

0

,

A

1

,

…

,

A

s

.

By the definitions of the matrices

A

i

and the second condition, for every pair

(

r

,

s

)

, exactly one of the

s

+

1

matrices has

1

in cell

(

r

,

s

)

, and all others have

0

in the corresponding cell. As a result, the

s

+

1

matrices are

linearly independent

.

Also, in view of the discussion above, it is easy to see that

A

i

⁢

A

j

=

p

i

⁢

j

0

⁢

A

0

+

p

i

⁢

j

1

⁢

A

1

+

⋯

+

p

i

⁢

j

s

⁢

A

s

.

Some terminology

.

s

is called the

rank

of the association scheme

𝒬

. Any

a

∈

Ω

, an element

c

∈

𝒬

is said to be an

i

-associate

of

a

if

(

a

,

c

)

∈

C

i

. For each

a

∈

Ω

, define:

C

i

⁢

(

a

)

:=

{

c

∈

Ω

∣

(

a

,

c

)

∈

C

i

}

.

So

C

i

⁢

(

a

)

is the set of all

i

-associates of

a

. Then

|

C

i

⁢

(

a

)

∩

C

j

⁢

(

b

)

|

=

p

i

⁢

j

k

,

whenever

⁢

(

a

,

b

)

∈

C

k

,

and because of the above

equation

, each

p

i

⁢

j

k

is called an

intersection number

of

𝒬

. For each

i

, the intersection number

p

i

⁢

i

0

is called the

valency

of

C

i

, denoted by

a

i

.

Some basic properties of the intersection numbers:

•

p

i

⁢

j

k

=

p

j

⁢

i

k

•

p

i

⁢

0

k

=

{

1

if

⁢

i

=

k

0

otherwise.

•

|

C

i

⁢

(

c

)

|

=

p

i

⁢

i

0

=

a

i

for all

c

∈

Ω

.

References

1

R. A. Bailey,

Association Schemes, Designed Experiments,

Algebra

and Combinatorics

, Cambridge University Press (2004)

Title

association scheme

Canonical name

AssociationScheme

Date of creation

2013-03-22 19:15:03

Last modified on

2013-03-22 19:15:03

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05E30

Defines

associate class

Defines

rank

Defines

intersection number

Defines

valency

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