AbstractA graph, with a group G of its automorphisms, is said to be (G,s)-transitive if G is transitive on s-arcs but not on (s+1)-arcs of the graph. Let X be a connected (G,s)-transitive graph for some s≥1, and let Gv be the stabilizer of a vertex v∈V(X) in G. In this paper, we determine the structure of Gv when X has valency 5 and Gv is non-solvable. Together with the results of Zhou and Feng [J.-X. Zhou, Y.-Q. Feng, On symmetric graphs of valency five, Discrete Math. 310 (2010) 1725–1732], the structure of Gv is completely determined when X has valency 5. For valency 3 or 4, the structure of Gv is known

Feng, Yan-Quan

Guo, Song-Tao

Elsevier - Publisher Connector 

Discrete Mathematics 312 (2012) 2214–2216
Contents lists available at SciVerse ScienceDirect
Discrete Mathematics
journal homepage: www.elsevier.com/locate/disc
Note
A note on pentavalent s-transitive graphs
Song-Tao Guo, Yan-Quan Feng ∗
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, PR China
a r t i c l e i n f o
Article history:
Received 22 April 2011
Received in revised form 28 February 2012
Accepted 18 April 2012
Available online 15 May 2012
Keywords:
Symmetric graph
s-arc-transitive graph
s-transitive graph
a b s t r a c t
A graph, with a group G of its automorphisms, is said to be (G, s)-transitive if G is transitive
on s-arcs but not on (s+ 1)-arcs of the graph. Let X be a connected (G, s)-transitive graph
for some s ≥ 1, and let Gv be the stabilizer of a vertex v ∈ V (X) in G. In this paper, we
determine the structure of Gv when X has valency 5 and Gv is non-solvable. Together with
the results of Zhou and Feng [J.-X. Zhou, Y.-Q. Feng, On symmetric graphs of valency five,
Discrete Math. 310 (2010) 1725–1732], the structure of Gv is completely determinedwhen
X has valency 5. For valency 3 or 4, the structure of Gv is known.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Throughout this paper, we consider undirected finite graphs without loops or multiple edges. For a graph X , we use
V (X), E(X) and Aut(X) to denote its vertex set, edge set, and its full automorphism group, respectively. Let G ≤ Aut(X)
and let S be a subset of V (X). Denote by G(S) the subgroup of G fixing S pointwise. In particular, for u, v, w ∈ V (X), write
Gu = G({u}),Guv = G({u,v}) and Guvw = G({u,v,w}). For u, v ∈ V (X), {u, v} is the edge incident to u and v in X , and X1(v) is the
neighborhood of v in X . Write G∗v = G({v}∪X1(v)) and G∗uv = G({u,v}∪X1(v)∪X1(u)).
An s-arc in a graph X is an ordered (s + 1)-tuple (v0, v1, . . . , vs−1, vs) of vertices of X such that vi−1 is adjacent to
vi for 1 ≤ i ≤ s, and vi−1 ≠ vi+1 for 1 ≤ i ≤ s − 1. A 1-arc is called an arc for short and a 0-arc is a vertex.
For a subgroup G ≤ Aut(X), X is said to be (G, s)-arc-transitive or (G, s)-regular if G is transitive or regular on the set
of s-arcs in X , respectively. An (G, s)-arc-transitive graph is said to be (G, s)-transitive if the graph is not (G, s + 1)-arc-
transitive. A graph X is called s-arc-transitive, s-regular or s-transitive if it is (Aut(X), s)-arc-transitive, (Aut(X), s)-regular
or (Aut(X), s)-transitive, respectively. In particular, X is said to be vertex-transitive or symmetric if it is (Aut(X), 0)-arc-
transitive or (Aut(X), 1)-arc-transitive, respectively.
Let X be a connected (G, s)-transitive graph for some s ≥ 1 and let Gv be the stabilizer of v ∈ V (X) in G. If X is cubic
then, by Tutte [6], G is s-regular for some 1 ≤ s ≤ 5, and by Djoković and Miller [2], Gv is isomorphic to Z3, S3, S3 × Z2, S4
or S4 × Z2 for s = 1, 2, 3, 4 or 5, respectively. If X has valency 4, then by [1], Gv is isomorphic to a 2-group for s = 1; by
[4, Theorem 4], Gv is isomorphic to A4 or S4 for s = 2 and to Z3×A4,Z3o S4 or S3× S4 for s = 3; by [12, Theorem 1.1], Gv is
isomorphic to Z23 o GL(2, 3) for s = 4, and to [35]o GL(2, 3) for s = 7. Let X be of valency 5. By Weiss [7,8], |Gv| is a divisor
of 217 · 32 · 5, and by Zhou and Feng [13], if Gv is solvable then Gv is isomorphic to Z5,D10 or D20 for s = 1, F20 or F20×Z2 for
s = 2, F20 × Z4 for s = 3. In this paper, we determine the structure of Gv when Gv is non-solvable and therefore we obtain
the following theorem.
Theorem 1.1. Let X be a connected pentavalent (G, s)-transitive graph for some G ≤ Aut(X) and s ≥ 1. Let v ∈ V (X). Then
s ≤ 5 and one of the following holds:
∗ Corresponding author.
E-mail addresses: gsongtao@gmail.com (S.-T. Guo), yqfeng@bjtu.edu.cn (Y.-Q. Feng).
0012-365X/$ – see front matter© 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.disc.2012.04.015
S.-T. Guo, Y.-Q. Feng / Discrete Mathematics 312 (2012) 2214–2216 2215
(1) For s = 1,Gv ∼= Z5,D10 or D20;
(2) For s = 2,Gv ∼= F20, F20 × Z2,A5 or S5;
(3) For s = 3,Gv ∼= F20 × Z4,A4 × A5, S4 × S5 or (A4 × A5) o Z2 with A4 o Z2 = S4 and A5 o Z2 = S5;
(4) For s = 4,Gv ∼= ASL(2, 4),AGL(2, 4),A6 L(2, 4) or A0 L(2, 4);
(5) For s = 5,Gv ∼= Z62 o 0L(2, 4).
Remark. For s = 5, by [12, Theorem 1.1] we have Gv = ⟨e0, e1, e2, e3, e4, c, f , g⟩ ∼= Z62 o 0L(2, 4) with the following
relations: e20 = e21 = e22 = e23 = e24 = c3 = f 3 = g2 = 1, [e0, e1] = [e0, e2] = [e1, e2] = [e1, e3] = [e2, e3] = [e2, e4] =
[e3, e4] = 1, [e0, e3] = e1e2, [e1, e4] = e2e3, (e0e4)3 = (e0c)3 = (e1f )3 = 1, (ce0e4)5 = 1, ce2 = c, ce0e4e0 = c−1, (cf )e0 =
cf , f e3 = f , (cf )e4 = cf , [c, f ] = 1, cg = c−1, f g = f −1, [ei, g] = 1 for all 0 ≤ i ≤ 4. By [10, Theorem 1.3], Gv has a
minimal normal subgroup Z22 such that Gv/Z
2
2
∼= A0 L(2, 4), and Gv is isomorphic to a maximal subgroup (isomorphic to
Z62 o ((Z3× A5)oZ2)) of PSp(4, 4)oZ2, where Z2 is the automorphism group of PSp(4, 4) induced by field automorphism
of GF(4).
2. Proof of Theorem 1.1
In this section, we prove Theorem 1.1. Let X be a connected pentavalent (G, s)-transitive graph for some G ≤ Aut(X) and
s ≥ 1. Let v ∈ V (X). Then s ≤ 5 by [11, Theorem]. If Gv is solvable then, by [13, Theorem 4.1], Gv is isomorphic to Z5,D10 or
D20 for s = 1, F20 or F20 × Z2 for s = 2, F20 × Z4 for s = 3. In what follows we assume that Gv is non-solvable.
Denote by GX1(v)v the constituent of Gv on X1(v), that is, the permutation group induced by Gv on X1(v). Since X is
pentavalent, GX1(v)v = Gv/G∗v ≤ S5. Let {u, v} be an edge in X and let S be a subset of V (X) such that u, v ∈ S and the
induced subgraph ⟨S⟩ of S in X is connected. Each vertex adjacent to S, but not in S, lies on an orbit of G(S) with length at
most 4 and then it is easy to see that Guv is a {2, 3}-group by an inductive method on |S|. By [5, Theorem 8.5.3], each finite
group of order pmqn with p, q primes is solvable and hence Guv is solvable. In particular, G∗v is solvable. Since Gv is non-
solvable, Gv/G∗v is non-solvable. Thus, G
X1(v)
v = Gv/G∗v = A5 or S5. This implies that s ≥ 2 and GX1(v)\{u}uv = A4 or S4. Note that
each non-trivial normal subgroup of A4 or S4 is transitive on X1(v) \ {u}. Since G∗u ▹ Guv,G∗u acts trivially or transitively on
X1(v) \ {u}.
Assume thatG∗u acts trivially on X1(v)\{u}. ThenG∗u ≤ G∗v and henceG∗u = G∗v because the transitivity ofG on V (X) implies|G∗u| = |G∗v|. Since G∗v E Gv and G∗u E Gu, we have G∗v E ⟨Gu,Gv⟩. By the transitivity of G on the set of arcs in X, ⟨Gu,Gv⟩ is
transitive on the set of edges in X , and since G∗v fixes the edge {u, v}, we may deduce that G∗v fixes each edge in X , forcing
that G∗v = 1. Thus, Gv = A5 or S5. In both cases, s = 2.
Assume that G∗u acts transitively on X1(v) \ {u}. Since s ≥ 2, we have s = 3, 4 or 5.
Let s = 3. By [3, Corollary 2.3], |G∗uv| = pt for some integer t and prime p. First we claim |G∗uv| = 1. Let (u, v, w, x) be a
3-arc in X . Suppose that G∗uv acts non-trivially on X1(w) \ {v}. Since (G∗uv)X1(w)\{v} E (G∗v)X1(w)\{v} E (Gvw)X1(w)\{v} = A4 or S4
and G∗uv is a p-group, we can deduce that (G∗uv)X1(w)\{v} is a non-trivial 2-group and (G∗uv)X1(w)\{v} ≤ A4. On the other hand,
GX1(w)\{v,x}vwx = Z3 or S3. Thus, Gvwx has an element g of order a power of 3 which induces a 3-cycle on X1(w)\{v, x}. Clearly, g
fixes at least two vertices in X1(v); one isw and since G is transitive on the set of 3-arcs in X , we may assume that the other
is u. Then g ∈ Guvwx induces a 3-cycle on X1(w)\{v}. Note that g normalizes G∗uv , that is, (G∗uv)g = G∗uv . Thus, G∗uv is transitive
on X1(w) \ {v} because (G∗uv)X1(w)\{v} ≤ A4 is a non-trivial 2-group. This implies that s ≥ 4, a contradiction. Thus, G∗uv acts
trivially on X1(w) \ {v}. In this case, G∗uv ≤ G∗vw and since G is transitive on the set of 3-arcs in X , we have |G∗uv| = |G∗vw|,
forcing G∗uv = G∗vw . By an inductive method on the distance from v to y, it is easy to see that G∗uv = G∗yz for any edge {y, z} in
X . It follows that G∗uv = 1, as claimed. Thus, G∗u acts transitively and faithfully on X1(v) \ {u}.
Set H = ⟨G∗z | z ∈ X1(v)⟩. Then H ▹ Gv . Recall that GX1(v)v = A5 or S5, and for each z ∈ X1(v),G∗z is transitive on
X1(v) \ {z}. It follows that H is transitive on X1(v), and hence HX1(v) = A5 or S5. Take h ∈ H such that h fixes uwith a 3-cycle
on X1(v) and has order a power of 3. Then h fixes some vertex y ∈ X1(u) with y ≠ v. Let x ∈ X1(v) and let α ∈ G∗v, β ∈ G∗z ,
where z is a fixed (but arbitrary) neighbor of v. Then xα
−1β−1αβ = xβ−1αβ = (xβ−1)αβ = (xβ−1)β = x and also this is true for
any x ∈ X1(z) because xα−1 ∈ X1(z). Thus, [G∗v,G∗z ] ≤ G∗vz = 1 and hence [G∗v,H] = 1. It follows that H ∩ G∗v ≤ Z(G∗v), the
center of G∗v . Note that h commutes with every element in G∗v . Since G∗u acts transitively on X1(v) \ {u}, the arc-transitivity
of G implies that G∗v is transitive on X1(u) \ {v}, and since h fixes y, h fixes every vertex in X1(u). Thus, h ∈ G∗u and h has a
3-cycle on X1(v). Since G∗u acts transitively and faithfully on X1(v) \ {u}, we have G∗u = S4 or A4.
By the transitivity of G on V (X),G∗v = S4 or A4. Thus, H ∩ G∗v ≤ Z(G∗v) = 1. This implies that H acts faithfully on X1(v)
and hence H = A5 or S5 because HX1(v) = A5 or S5. Furthermore, G∗vH = G∗v × H .
Now we deal with the two cases: G∗v = S4 or A4. If G∗v = S4 then for each z ∈ X1(v), (G∗z )X1(v)\{z} = S4. This implies that
HX1(v) = S5 because H = ⟨G∗z | z ∈ X1(v)⟩, and since H is faithful on X1(v), we have H = S5. Similarly, if G∗v = A4 then
H = A5. Since Gv/G∗v = A5 or S5, we have Gv = G∗v × H or |Gv : G∗v × H| = 2. For the former, Gv = S4 × S5 or A4 × A5. For
the latter, G∗v = A4 and H = A5. In this case, Gv = (A4 × A5).Z2, and to finish the proof for s = 3, it suffices to show that
Gv = (A4 × A5) o Z2 with A4 o Z2 = S4 and A5 o Z2 = S5, which is equivalent to finding an involution g ∈ Gv such that
H⟨g⟩ = S5 and G∗v⟨g⟩ = S4.
2216 S.-T. Guo, Y.-Q. Feng / Discrete Mathematics 312 (2012) 2214–2216
Since |Gv : G∗vH| = 2, we have G∗vH = A4 × A5 ▹ Gv . Recall that G∗v ▹ Gv and H ▹ Gv . Since GX1(v)v = Gv/G∗v = S5, there
exists g1 ∈ Guv such that g1 induces a transposition on X1(v). Since G∗v = A4 and G∗v acts faithfully on X1(u), there is g2 ∈ G∗v
such that g1g2 induces the identity or a transposition on X1(u) \ {v}. For the former, g1g2 ∈ G∗u and g1g2 induces the same
transposition as g1 on X1(v), contrary to the fact that G∗u = A4, acting faithfully on X1(v). Set g = g1g2. Thus, g ∈ Guv induces
transposition on both X1(u) and X1(v). Furthermore, g2 ∈ G∗uv = 1 and hence g is an involution. It follows that H⟨g⟩ = S5
and G∗v⟨g⟩ = S4 because G∗v ▹ Gv and H ▹ Gv , as required.
For s = 4, by [12, Theorems 1.1 and 1.2] we have Z42 o SL(2, 4) ≤ Gv ≤ Z42 o 0L(2, 4) and Z42 o 0L(2, 4) = ⟨e0,
e1, e2, e3, c, f , g⟩, where e20 = e21 = e22 = e23 = c3 = f 3 = g2 = 1, [e0, e1] = [e0, e2] = [e1, e2] = [e2, e3] = 1, [e1, e3] =
e2, ce0e3e0 = c−1, (e0e3)3 = (e0c)3 = (ce1)3 = (ce2)3 = 1, (ce0e3)5 = 1, [e0, f ] = 1, f e1 = e1ec1f , f e2 = e2ec2f , [e3, f ] =
1, [e0, g] = [e1, g] = [e2, g] = [e3, g] = 1, [c, f ] = 1, cg = c−1, f g = f . By [10, Theorem 1.3], Z42 o 0L(2, 4) = A0 L(2, 4)
and hence ASL(2, 4) ≤ Gv ≤ A0 L(2, 4). It follows that Gv = ASL(2, 4),AGL(2, 4),A6 L(2, 4) or A0 L(2, 4). For s = 5, again
by [12, Theorems 1.1 and 1.2], we have Gv ∼= Z62 o 0L(2, 4). This completes the proof of Theorem 1.1. 
3. Examples
Let X be a connected pentavalent (G, s)-transitive graph and let v ∈ V (X). By [13, Section 3], each solvable type of
Gv in Theorem 1.1 can be realized. In this section, we show that each non-solvable type of Gv in Theorem 1.1 can also be
realized, that is, there exists a graph X such that Gv ∼= A5, S5,A4 × A5, S4 × S5, (A4 × A5) o Z2 with A4 o Z2 = S4 and
A5 o Z2 = S5,ASL(2, 4),AGL(2, 4),A6 L(2, 4),A0 L(2, 4) or Z62 o 0L(2, 4), respectively.
Example 3.1. Let X = K6 be the complete graph of order 6. Then A = Aut(X) = S6. Clearly, A has an arc-transitive subgroup
B isomorphic to A6. Thus, the vertex stabilizers Av and Bv of v ∈ V (K6) in A and B are isomorphic to S5 and A5, respectively.
The following example is a pentavalent (G, 3)-transitive graph with Gv isomorphic to S4× S5,A4× A5 or (A4× A5)oZ2
with A4 o Z2 = S4 and A5 o Z2 = S5.
Example 3.2. Let X = K5,5 be the complete bipartite graph of order 10 with bipartite sets {1, 3, 5, 7, 9} and {2, 4, 6, 8, 10}.
Then A = Aut(X) ∼= S5wr S2. Clearly, A has a 3-transitive subgroup B isomorphic to A5wr S2. Thus, the vertex stabilizers
A1 and B1 of the vertex 1 in A and B are isomorphic to S4 × S5 and A4 × A5, respectively. Furthermore, let C = ⟨B, a⟩ with
a = (5 7)(6 8). Then C is 3-transitive and C1 ∼= (A4 × A5) o Z2 with A4 o Z2 = S4 and A5 o Z2 = S5.
In what follows, we give two classical generalized polygon graphs (see [9,10]). The first is a 4-transitive graph.
Example 3.3. Let X be the classical generalized 4-gon graph related to the group PSL(3, 4). Then A = Aut(X) ∼=
Aut(PSL(3, 4)). Since Out(PSL(3, 4)) ∼= D12, we may assume that Aut(PSL(3, 4)) = T o L with T = Inn(PSL(3, 4)) and
L = ⟨a, b | a2 = b6 = 1⟩. Write B = ⟨T , a, b3⟩, C = ⟨T , b⟩, and D = T o ⟨b3⟩. Then A, B, C and D are all 4-transitive with
vertex stabilizers isomorphic to A0 L(2, 4),A6 L(2, 4),AGL(2, 4) and ASL(2, 4), respectively.
At last we give a pentavalent (G, 5)-transitive graph whose vertex stabilizer Gv is isomorphic to Z62 o 0L(2, 4).
Example 3.4. Let X be the classical generalized 4-gon graph related to the group PSp(4, 4). Then Aut(X) ∼= Aut(PSp(4, 4))
with a vertex stabilizer isomorphic to Z62 o 0L(2, 4).
Acknowledgments
The authors are grateful to R.M.Weiss for his helpful comments. Thisworkwas supported by theNational Natural Science
Foundation of China (11171020).
References
[1] D.Ž Djoković, A class of finite group-amalgams, Proc. Amer. Math. Soc. 80 (1980) 22–26.
[2] D.Ž Djoković, G.L. Miller, Regular groups of automorphisms of cubic graphs, J. Combin. Theory Ser. B 29 (1980) 195–230.
[3] A. Gardiner, Arc transitivity in graphs, Quart. J. Math. Oxford 24 (1973) 399–407.
[4] P. Potočnik, A list of 4-valent 2-arc-transitive graphs and finite faithful amalgams of index (4, 2), Electron. J. Combin. 30 (2009) 1323–1336.
[5] D.J. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1982.
[6] W.T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947) 621–624.
[7] R.M. Weiss, Über symmetrische Graphen vom Grad fünf, J. Combin. Theory Ser. B 17 (1974) 59–64.
[8] R.M. Weiss, Über symmetrische graphen deren valenz eine primzahl ist, Math. Z. 136 (1974) 277–278.
[9] R.M. Weiss, Über Tuttes Cages, Monatsh. Math. 83 (1977) 65–75.
[10] R.M. Weiss, Groups with a (B,N)-pair and locally transitive graphs, Nagoya Math. J. 74 (1979) 1–21.
[11] R.M. Weiss, The nonexistence of 8-transitive graphs, Combinatorica 1 (1981) 309–311.
[12] R.M. Weiss, Presentations for (G, s)-transitive graphs of small valency, Math. Proc. Cambridge Philos. Soc. 101 (1987) 7–20.
[13] J.-X. Zhou, Y.-Q. Feng, On symmetric graphs of valency five, Discrete Math. 310 (2010) 1725–1732.


A note on pentavalent s-transitive graphs 

file:///data/remote/core/dit/data/elsevier/pdf/dfb/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAxMjM2NXgxMjAwMTkwMg==.pdf

A note on pentavalent s-transitive graphs

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector