Moore graph

Moore graph

\PMlinkescapephrase

one way

Moore graphs

It is easy to see that if a graph

G

has

diameter

d

(and has any cycles
at all), its girth

g

can be no more than

2

⁢

d

+

1

. For suppose

g

⩾

2

⁢

d

+

2

and

O

is a cycle of that minimum length

g

. Take two vertices (nodes) A and B
on

O

that are

d

+

1

steps apart along

O

, one way round; they are

g

-

(

d

+

1

)

⩾

d

+

1

steps apart the other way round. Now either there is no shorter
route between A and B (contradicting diameter

d

) or there is a shorter route
of length

r

<

d

+

1

creating a cycle of length

d

+

1

+

r

<

2

⁢

d

+

2

(contradicting girth

g

).

Definition

: A

Moore graph

is a

connected graph

with

maximal

girth

2

⁢

d

+

1

for its diameter

d

.

It can be shown that Moore graphs are

regular

, i.e. all vertices have the
same

valency

. So a Moore graph is characterised by its diameter and valency.

Moore graphs with

d

=

1

,

g

=

3

Diameter 1 means every vertex (node) is

adjacent

to every other, that is, a

complete graph

. Indeed, complete graphs

K

n

that have cycles (

n

⩾

3

) have

triangles

, so the girth is 3 and they are Moore graphs. Every valency

v

⩾

2

occurs (

K

n

has valency

n

-

1

).

Moore graphs with

d

=

2

,

g

=

5

This is the most interesting case. And

http://planetmath.org/node/6948

the proof that every vertex has the same valency,

v

say, and that the graph now has

n

=

v

2

+

1

vertices in all, is easy here.

With some more work, it can be shown there are only 4 possible values for

v

and

n

:

v

n

graph

2

5

pentagon

3

10

Petersen graph

7

50

Hoffman–

Singleton graph

57

3250

???

The first three cases each have a unique

solution

. The existence or otherwise of
the last case is still

http://planetmath.org/node/6331

open. It has been shown that if it exists it has, unlike the first three, very little

symmetry

.

The Hoffman–Singleton graph is a bit hard to draw. Here’s a unified
description of the three known Moore graphs of

d

=

2

, all indices (mod 5):

•

Pentagon:

five vertices, which we can label

P

k

.

Each

P

k

is connected by two edges
to

P

k

±

1

.

•

Petersen:

ten vertices, which we can label

P

k

∘

and

P

k

⋆

.

Each

P

k

∘

is connected by two edges to

P

k

±

1

∘

and by one to

P

2

⁢

k

⋆

.

Each

P

k

⋆

is connected by two edges to

P

k

±

1

⋆

and by one to

P

3

⁢

k

∘

.

•

Hoffman–Singleton:

fifty vertices, which we can label

P

n

,

k

∘

and

P

m

,

k

⋆

.

Each

P

n

,

k

∘

is connected by two edges
to

P

n

,

k

±

1

∘

and by five to

P

m

,

m

⁢

n

+

2

⁢

k

⋆

for all

m

.

Each

P

m

,

k

⋆

is connected by two edges
to

P

m

,

k

±

1

⋆

and by five to

P

n

,

2

⁢

m

⁢

n

+

3

⁢

k

∘

for all

n

.

The

automorphism group

of the pentagon is the

dihedral group

with 10 elements. The one of the Petersen graph is

isomorphic

to

S

5

, with 120 elements. And the one of the HS graph is isomorphic to

PSU

⁢

(

3

,

5

)

⋅

2

, with 252 000 elements, a maximal subgroup of another HS, the

Higman-Sims group

.

Moore graphs with

d

⩾

3

,

g

⩾

7

In these cases, there are only Moore graphs with valency 2, graphs
consisting of a single

2

⁢

d

+

1

-gon cycle. This was proven independently by
Bannai and Ito and by Damerell.

Title

Moore graph

Canonical name

MooreGraph

Date of creation

2013-03-22 15:11:27

Last modified on

2013-03-22 15:11:27

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05C75

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