A connected graph Γ=(V,E) is called a basic 2-arc-transitive graph if its full automorphism group has a 2-arc-transitive subgroup G, and every minimal normal subgroup of G has at most two orbits on ...V. In 1993, Praeger proved that every finite 2-arc-transitive connected graph is a cover of some basic 2-arc-transitive graph, and proposed the classification problem of finite basic 2-arc-transitive graphs. In this paper, a classification is given for basic 2-arc-transitive non-bipartite graphs of order rasb and basic 2-arc-transitive bipartite graphs of order 2rasb, where r and s are distinct primes.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
A vertex triple (u,v,w) of a graph is called a 2-geodesic if v is adjacent to both u and w and u is not adjacent to w. A graph is said to be 2-geodesic transitive if its automorphism group is ...transitive on the set of 2-geodesics. In this paper, a complete classification of 2-geodesic transitive graphs of order pn is given for each prime p and n≤3. It turns out that all such graphs consist of three small graphs: the complete bipartite graph K4,4 of order 8, the Schläfli graph of order 27 and its complement, and fourteen infinite families: the cycles Cp,Cp2 and Cp3, the complete graphs Kp,Kp2 and Kp3, the complete multipartite graphs Kpp, Kpp2 and Kp2p, the Hamming graph H(2,p) and its complement, the Hamming graph H(3,p), and two infinite families of normal Cayley graphs on the extraspecial group of order p3 and exponent p.
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A plenty of contributions have been done on symmetric Cayley graphs on nonabelian simple groups, but the only known complete classification of such graphs with composite valency is of valency 4 ...(provided 2-arc-transitivity) by Fang et al. Europ. J. Combin. 25 (2004), 1107-1116 and Du and Feng Comm. Algebra 47 (2019), 4565-4574. This naturally motivates this work for classifying 2-arc-transitive hexavalent Cayley graphs on nonabelian simple groups. It is proved that these graphs are either normal or
-arc-transitive Cayley graphs on
where n is among 11 specific numbers dividing
A specific example is also constructed.
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BFBNIB, GIS, IJS, KISLJ, NUK, PNG, UL, UM, UPUK
A connected graph
Γ
=
(
V
,
E
)
of valency at least 3 is called a basic 2-arc-transitive graph if its full automorphism group has a subgroup
G
with the following properties: (i)
G
acts transitively ...on the set of 2-arcs of
Γ
, and (ii) every minimal normal subgroup of
G
has at most two orbits on
V
. Based on Praeger’s theorems on 2-arc-transitive graphs, this paper presents a further understanding on the automorphism group of a basic 2-arc-transitive graph.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
A graph is edge-primitive if its automorphism group acts primitively on the edge set. In this short paper, we prove that a finite 2-arc-transitive edge-primitive graph has almost simple automorphism ...group if it is neither a cycle nor a complete bipartite graph. We also present two examples of such graphs, which are 3-arc-transitive and have faithful vertex-stabilizers.
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We classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that either Γ is 2‐arc ...transitive or the valency p satisfies p≡1(mod4), and for each such prime there is a unique graph with this property: it is a nonbipartite antipodal double cover of the complete graph Kp+1 with automorphism group PSL(2,p)×Z2 and diameter 3.
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Finite 2‐Distance Transitive Graphs Corr, Brian P.; Jin, Wei; Schneider, Csaba
Journal of graph theory,
September 2017, 2017-09-00, 20170901, Volume:
86, Issue:
1
Journal Article
Peer reviewed
Open access
A noncomplete graph Γ is said to be (G, 2)‐distance transitive if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set of Γ, and for any vertex u of Γ, the stabilizer ...Gu is transitive on the sets of vertices at distances 1 and 2 from u. This article investigates the family of (G, 2)‐distance transitive graphs that are not (G, 2)‐arc transitive. Our main result is the classification of such graphs of valency not greater than 5. We also prove several results about (G, 2)‐distance transitive, but not (G, 2)‐arc transitive graphs of girth 4.
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Classification of 2-arc-transitive dihedrants Du, Shaofei; Malnič, Aleksander; Marušič, Dragan
Journal of combinatorial theory. Series B,
11/2008, Volume:
98, Issue:
6
Journal Article
Peer reviewed
Open access
A complete classification of 2-arc-transitive
dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in D. Marušič, On ...2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162–196. The list consists of the following graphs:
(i)
cycles
C
2
n
,
n
⩾
3
;
(ii)
complete graphs
K
2
n
,
n
⩾
3
;
(iii)
complete bipartite graphs
K
n
,
n
,
n
⩾
3
;
(iv)
complete bipartite graphs minus a matching
K
n
,
n
−
n
K
2
,
n
⩾
3
;
(v)
incidence and nonincidence graphs
B
(
H
11
)
and
B
′
(
H
11
)
of the Hadamard design on 11 points;
(vi)
incidence and nonincidence graphs
B
(
PG
(
d
,
q
)
)
and
B
′
(
PG
(
d
,
q
)
)
, with
d
⩾
2
and
q a prime power, of projective spaces;
(vii)
and an infinite family of regular
Z
d
-covers
K
q
+
1
2
d
of
K
q
+
1
,
q
+
1
−
(
q
+
1
)
K
2
, where
q
⩾
3
is an odd prime power and
d is a divisor of
q
−
1
2
and
q
−
1
, respectively, depending on whether
q
≡
1
(
mod
4
)
or
q
≡
3
(
mod
4
)
, obtained by identifying the vertex set of the base graph with two copies of the projective line
PG
(
1
,
q
)
, where the missing matching consists of all pairs of the form
i
,
i
′
,
i
∈
PG
(
1
,
q
)
, and the edge
i
,
j
′
carries trivial voltage if
i
=
∞
or
j
=
∞
, and carries voltage
h
¯
∈
Z
d
, the residue class of
h
∈
Z
, if and only if
i
−
j
=
θ
h
, where
θ generates the multiplicative group
F
q
∗
of the Galois field
F
q
.
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Locally 2-arc-transitive complete bipartite graphs Fan, Wenwen; Leemans, Dimitri; Li, Cai Heng ...
Journal of combinatorial theory. Series A,
April 2013, 2013-04-00, Volume:
120, Issue:
3
Journal Article
Peer reviewed
Open access
We study finite groups that act edge-transitively on a complete bipartite graph. In particular, we obtain a classification of the groups of automorphisms for the locally 2-arc-transitive case.
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