ISSN 10645624, Doklady Mathematics, 2013, Vol. 87, No. 1, pp. 31–33. © Pleiades Publishing, Ltd., 2013.
Original Russian Text © K.V. Kostousov, 2013, published in Doklady Akademii Nauk, 2013, Vol. 448, No. 2, pp. 136–138.
31
Throughout the paper, by a graph we mean an
undirected graph without loops or multiple edges. The
vertex set and edge set of a graph
Γ
are denoted by
V
(
Γ
)
and
E
(
Γ
)
. By an automorphism of graph
Γ
we mean a
permutation on the set
V
(
Γ
)
preserving the adjacency
relation.
A graph is called a vertexprimitive graph if it
admits a primitive on its vertex set group of automor
phisms. We denote the class of connected vertexprim
itive graphs by (here and below by a class of some
graphs we mean a set of isomorphic types of these
graphs). Each primitive permutation group can be
realized as an edgetransitive automorphism group of
some connected graph. Here the most natural realiza
tions are obtained via graphs of minimal degree. A
connected graph is called a
graph of minimal
degree for
a primitive permutation group of automorphisms
G
on
the set
V
if it has a minimal degree among all con
nected graphs
Γ
, with
V
(
Γ
) =
V
and
G
≤
Aut(
Γ
)
. The
subclass of the class
consisting of all graphs of
minimal degree for finite primitive permutation
groups is denoted by .
For investigation of class, it was proposed an
approach in [1, 3] connected with investigation of
limit graphs for class. If
C
is an arbitrary class of
connected vertexprimitive graphs, then an infinite
connected graph, each ball of which is isomorphic to a
ball of some graph from
C
, is called a
limit graph
for
C
.
The class of limit graphs for
C
is denoted by
lim(
C
)
. A
description of
lim(
C
)
provides a useful description of
the possible local structures of generic graphs from
C
.
Ᏺᏼ
Ᏺᏼ
min
Ᏺᏼ
Ᏺᏼ
Ᏺᏼ
The problem of description of graphs from
was posed by Trofimov in [2]. Mentioned above
implies that investigation of the structure of graphs
from is also of interest.
The systematic study of class was started
by Giudici et al. in [3]. In this connection it was shown
that
where the subclasses
, ,
of class
are defined as follows. Let
G
be a primitive permuta
tion group acting on the set
V
and
v
∈
V
. If
G
has an
abelian normal regular subgroup, then
G
is said to be of
type HA. If
G
is an almost simple group, i.e., there
exists a finite nonabelian simple group
T
such that
Inn(
T
)
G
≤
Aut(
T
)
, then
G
is said to be of type AS.
If
G
has a minimal normal subgroup
N = T
k
(
k
≥
2
) for
some nonabelian simple group
T
and the stabiliser in
N
of
v
is nontrivial and has no composition factor iso
morphic to
T
, then
G
is said to be of type PA. For each
primitive type
X
, the class of all vertexprimitive graphs
with an automorphism group of type
X
is denoted by
, and the subclass of consisting of all
graphs of minimal degree for vertexprimitive groups
of type
X
is denoted by .
It follows from [3, Theorem 1] that
Therefore the investigation of class, which
is of independent interest, is a necessary stage of the
investigation of class.
In [4–7], all graphs of degree
≤
14
from
(12 graphs) were found, and also a countable set of set
wisenonisomophic graphs of degree 24 from
lim( )
Ᏺᏼ
min
lim( )
Ᏺᏼ
lim( )
Ᏺᏼ
=∪∪
HA AS PA
lim( ) lim( ) lim( ) lim( ),
Ᏺᏼ Ᏺᏼ Ᏺᏼ Ᏺᏼ
HA
Ᏺᏼ
AS
Ᏺᏼ
PA
Ᏺᏼ
Ᏺᏼ
䉰
X
Ᏺᏼ
X
min
Ᏺᏼ
X
Ᏺᏼ
HA AS PA
min
min min min
lim( )
lim( ) lim( ) lim( ).
=∪∪
Ᏺᏼ
Ᏺᏼ Ᏺᏼ Ᏺᏼ
HA
min
lim( )
Ᏺᏼ
min
lim( )
Ᏺᏼ
HA
min
lim( )
Ᏺᏼ
Graphs of Degree <24, which are Limit Graphs
for Minimal VertexPrimitive Graphs of Type HA
1
K. V. Kostousov
Presented by Academician V.A. Il’in June 14, 2012
Received June 14, 2012
DOI:
10.1134/S1064562413010122
Institute of Mathematics and Mechanics (IMM),
Ural Branch of the Russian Academy of Sciences,
Yekaterinburg, Russia
Ural Federal University named after the first President
of Russia B.N. Yeltsin, Yekaterinburg, Russia
MATHEMATICS
1
The article was translated by the author.
32
DOKLADY MATHEMATICS Vol. 87 No. 1 2013
KOSTOUSOV
was found. In the present paper, we find all
graphs of degree <24 from (23 graphs, see
Theorem 1 below).
Let
d
be a positive integer. Let
M
be a set of gener
ators of
⺪
d
such that
M=
–
M
and
0
∉
M
. Recall that
a graph
⺪
d
,
M
with vertex set
⺪
d
and the set of edges
such that two vertexes are connected by an edge iff
their difference lies in
M=
–
M
and
0
∉
M
is called a
Caley graph of the group
⺪
d
corresponding to the set
of generators
M
. Let be a stabilizer of ver
tex 0 in automorphism group of . We say a graph
to be minimal Caley graph of
⺪
d
if
M
is an orbit
of minimal cardinality of group , acting on
the set . The class of all Cayley graphs of the
group
⺪
d
is denoted by , and the class of all
minimal Caley graphs of
⺪
d
is denoted by .
It follows from [3, Theorem 2] that each element of
lies in for some
d
. Morever, it fol
lows from [3, Theorem 2] and the definition of a limit
graphs that each element of lies in
for some
d
. In other words,
⊆
. For each , we identify
⺪
d
with a
set of integer row vectors of length
d
with coordinate
wise addition. For
i
∈
{1, 2, …,
d
}
, let
e
i
be a rowvec
tor of length
d
having 1 at position
i
and 0 at other
positions. Next we set
M
d,
1
= {
±
e
i
:
i
∈
{1, 2,
…
,
d
}}
and
.
Theorem 1.
The class of all graphs from
of degree
<24
is equal to the class of all graphs from
of degree
<24
and consists of following
graphs:
of degree
2;
of degree
4;
and
of degree
6;
of degree
8;
and
of degree
10;
HA
min
lim( )
Ᏺᏼ
HA
min
lim( )
Ᏺᏼ
Γ
Aut
0
,
()
d
M⺪
Γ
,
d
M
⺪
Γ
,
d
M
⺪
Γ
Aut
0
,
()
d
M⺪
\
{0}
d
⺪
Cay
()
d
⺪
Cay
min
()
d
⺪
HA
lim( )
Ᏺᏼ
Cay
()
d
⺪
HA
min
lim( )
Ᏺᏼ
Cay
min
()
d
⺪
HA
min
lim( )
Ᏺᏼ
∞
=
∪
Cay
min
1
()
d
d
⺪
≥
∈
0
i
⺪
21,,
1
d
dd i
i
MM e
=
⎧⎫
⎪⎪
=∪±
⎨⎬
⎪⎪
⎩⎭
∑
HA
min
lim( )
Ᏺᏼ
∞
=
∪
Cay
min
1
()
d
d
⺪
Γ
1,1
,M⺪
Γ
2
2,1
,M⺪
Γ
2
2,2
,M⺪
Γ
3
3,1
,M⺪
Γ
4
4,1
,M⺪
Γ
4
4,2
,M⺪
Γ
5
5,1
,M⺪
,
,
and
of degree
12,
where
M
4, 3
=
;
and
of degree
14;
and
of degree
16;
,
,
and
of degree
18,
where
M
6, 3
=
;
,
,
,
and
of degree
20,
where
M
6, 4
=
M
6, 1
∪ ±
{
e
1
–
e
2
+
e
4
,
e
1
–
e
3
+
e
5
,
e
2
–
e
3
+
e
6
,
e
4
–
e
5
+
e
6
}
,
;
and
of degree
22.
For each graph , the group is a
subgroup of , and acts naturally on
. Besides that, the group
induces a permutation group on the set of pairs of
mutually inverse vectors from the set
M
. The condi
tions of Theorem 1 imply . To prove Theorem 1,
for each minimal transitive permutation group
S
of
degree
≤
12
, we find all systems generator systems
М
of
⺪
d
, which are
G
orbits on of minimal cardi
nality for some subgroup
G
of , such that
G
induces on a permutationaly isomorphic to
S
group. Herewith we search for systems
M
up to the fol
lowing equivalency, which implies isomorphism of the
corresponding Caley graphs. Two systems of genera
tors
М
1
and
M
2
of a group
⺪
d
are equivalent if
M
2
=
М
1
A
for some . As a result, we get all
graphs from of degree <24. To prove
that each found in this way graph is contained in
, we construct an infinity set of graphs
,
from the class , where
p
i
is a
prime number,
p
i + 1
>
p
i
,
is a
d
dimensional space
of the residues modulo
p
i
, and
:
is a func
tion that substitutes integer elements of row vectors
my their residues modulo
p
i
. In Theorem 1 proof, we
use the classification of maximal finite subgroups of
group for
d
≤
11
and the classification of tran
sitive permutation groups for small degrees (both clas
sifications are available in [8]). Also we use earlier
results from [4–7] about the structure of class
for
d
≤
7
.
Γ
4
4,3
,M⺪
Γ
5
5,2
,M⺪
Γ
6
6,1
,M⺪
∪± + + +
4,1 1 2 3 4
{}
Meeee
Γ
6
6,2
,M⺪
Γ
7
7,1
,M⺪
Γ
7
7,2
,M⺪
Γ
8
8,1
,M⺪
Γ
6
6,3
,M⺪
Γ
8
8,2
,M⺪
Γ
9
9,1
,M⺪
∪± + + +
6,1 1 2 3 4 5 6
{, , }
Meeeeee
Γ
6
6,4
,M⺪
Γ
8
8,3
,M⺪
Γ
9
9,2
,M⺪
10
10,1
,M
Γ
⺪
=∪±+++ +++
8,3 8,1 1 2 3 4 5 6 7 8
{, }
M M eeeeeeee
Γ
10
10,2
,M⺪
Γ
11
11,1
,M⺪
Γ
,
d
M
⺪
Γ
Aut
0
,
()
d
M⺪
GL
()
d
⺪
=Γ
,
()
d
d
M
V
⺪
⺪
Γ
Aut
0
,
()
d
M⺪
M
≤
12
M
\
{0}
d
⺪
Γ
Aut
0
,
()
d
M⺪
M
∈
GL
()
d
A
⺡
∞
=
∪
Cay
min
1
()
d
d
⺪
Γ
,
d
M
⺪
HA
min
lim( )
Ᏺᏼ
ϕ
Γ
,()
d
pp
ii
M
⺪
≥
∈
0
i
⺪
HA
min
Ᏺᏼ
i
d
p
⺪
ϕ
i
p
→
i
dd
p
⺪⺪
GL
()
d
⺪
∩
HA
Cay
min min
lim( ) ( )
d
Ᏺᏼ
⺪
DOKLADY MATHEMATICS Vol. 87 No. 1 2013
GRAPHS OF DEGREE <24 33
ACKNOWLEDGMENTS
This work was supported by the Russian Founda
tion for Basic Research, project no. 100100349.
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