Arc-Transitive Graphs of Square-free Order with Valency 11

2021 ◽  
Vol 28 (04) ◽  
pp. 645-654
Author(s):  
Guang Li ◽  
Bo Ling ◽  
Zaiping Lu
Keyword(s):  
Simple Group ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Cayley Graphs ◽  
Complete List ◽  
Dihedral Groups ◽  
Transitive Graphs ◽  
Free Order ◽  
Normal Cayley Graphs

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

THE TWO-ARC-TRANSITIVE GRAPHS OF SQUARE-FREE ORDER ADMITTING ALTERNATING OR SYMMETRIC GROUPS

2017 ◽  
Vol 104 (1) ◽  
pp. 127-144
Author(s):  
GAI XIA WANG ◽  
ZAI PING LU
Keyword(s):  
Finite Group ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Symmetric Groups ◽  
Transitive Graph ◽  
A Finite Group ◽  
Transitive Graphs ◽  
Free Order

Let $G$ be a finite group with $\mathsf{soc}(G)=\text{A}_{c}$ for $c\geq 5$. A characterization of the subgroups with square-free index in $G$ is given. Also, it is shown that a $(G,2)$-arc-transitive graph of square-free order is isomorphic to a complete graph, a complete bipartite graph with a matching deleted or one of $11$ other graphs.

s-Arc-regular prime-valent Cayley graphs of square-free order

2019 ◽  
Vol 18 (05) ◽  
pp. 1950092
Author(s):  
Jiangmin Pan ◽  
Menglin Ding ◽  
Bo Ling
Keyword(s):  
Cayley Graphs ◽  
Complete List ◽  
Regular Graphs ◽  
Free Order

A high cited paper [Feng and Li, One-regular graphs of square-free order of prime valency, Europ. J. Combin. 32 (2011) 261–275] classified 1-arc-regular prime-valent graphs (necessarily Cayley graphs) of square-free order. It is thus interesting to ask what are the [Formula: see text]-arc-regular prime-valent Cayley graphs of square-free order for each [Formula: see text]. In this paper, a complete list of all such graphs is given. The proving process also involves a complete list of the vertex stabilizers of [Formula: see text]-arc-regular prime-valent graphs of any order.

AUTOMORPHISM GROUPS OF SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS

2015 ◽  
Vol 93 (2) ◽  
pp. 238-247
Author(s):  
ZHAOHONG HUANG ◽  
JIANGMIN PAN ◽  
SUYUN DING ◽  
ZHE LIU
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Automorphism Groups ◽  
Composition Factor ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs ◽  
Free Order

Li et al. [‘On finite self-complementary metacirculants’, J. Algebraic Combin.40 (2014), 1135–1144] proved that the automorphism group of a self-complementary metacirculant is either soluble or has $\text{A}_{5}$ as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of $4$-power-free order.

scholarly journals On Arc-Transitive Metacyclic Covers of Graphs with Order Twice a Prime

10.37236/7146 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Zhaohong Huang ◽  
Jiangmin Pan
Keyword(s):  
Bipartite Graph ◽  
Prime Order ◽  
Complete Bipartite Graph ◽  
Infinite Family ◽  
Petersen Graph ◽  
Regular Graphs ◽  
Dihedral Groups ◽  
Hadamard Design ◽  
Symmetric Graphs ◽  
Abelian Covers

Quite a lot of attention has been paid recently to the characterization and construction of edge- or arc-transitive abelian (mostly cyclic or elementary abelian) covers of symmetric graphs, but there are rare results for nonabelian covers since the voltage graph techniques are generally not easy to be used in this case. In this paper, we will classify certain metacyclic arc-transitive covers of all non-complete symmetric graphs with prime valency and twice a prime order $2p$ (involving the complete bipartite graph ${\sf K}_{p,p}$, the Petersen graph, the Heawood graph, the Hadamard design on $22$ points and an infinite family of prime-valent arc-regular graphs of dihedral groups). A few previous results are extended.

Pentavalent One-regular Graphs of Square-free Order

2010 ◽  
Vol 17 (03) ◽  
pp. 515-524 ◽  
Author(s):  
Yantao Li ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Regular Graph ◽  
Cayley Graphs ◽  
Regular Graphs ◽  
Dihedral Groups ◽  
Free Order

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. It is shown in this paper that a pentavalent one-regular graph of order n exists if and only if n = 2 · 5tp1p2 … ps ≥ 62, where t ≤ 1, s ≥ 1, and pi's are distinct primes such that 5|(pi-1). For such an integer n, there are exactly 4s-1 non-isomorphic pentavalent one-regular graphs of order n, which are Cayley graphs on dihedral groups constructed by Kwak et al. This work is a continuation of the classification of cubic one-regular graphs of order twice a square-free integer given by Zhou and Feng.

Isomorphisms of bi-Cayley graphs on Dihedral groups

2020 ◽  
Vol 12 (04) ◽  
pp. 2050051
Author(s):  
Majid Arezoomand ◽  
Afshin Behmaram ◽  
Mohsen Ghasemi ◽  
Parivash Raeighasht
Keyword(s):  
Bipartite Graph ◽  
Cayley Graph ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Dihedral Groups ◽  
Vertex Set

For a group [Formula: see text] and a subset [Formula: see text] of [Formula: see text] the bi-Cayley graph BCay[Formula: see text] of [Formula: see text] with respect to [Formula: see text] is the bipartite graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bi-Cayley graph BCay[Formula: see text] is called a BCI-graph if for any bi-Cayley graph BCay[Formula: see text], [Formula: see text] implies that [Formula: see text] for some [Formula: see text] and [Formula: see text]. A group [Formula: see text] is called a [Formula: see text]-BCI-group if all bi-Cayley graphs of [Formula: see text] with valency at most [Formula: see text] are BCI-graphs. In this paper, we characterize the [Formula: see text]-BCI dihedral groups for [Formula: see text]. Also, we show that the dihedral group [Formula: see text] ([Formula: see text] is prime) is a [Formula: see text]-BCI-group.

scholarly journals TETRAVALENT s-TRANSITIVE GRAPHS OF ORDER TWICE A PRIME POWER

2010 ◽  
Vol 88 (2) ◽  
pp. 277-288 ◽  
Author(s):  
JIN-XIN ZHOU ◽  
YAN-QUAN FENG
Keyword(s):  
Automorphism Group ◽  
Bipartite Graph ◽  
Projective Geometry ◽  
Prime Power ◽  
Complete Bipartite Graph ◽  
Three Dimensional ◽  
Transitive Graph ◽  
Incidence Graph ◽  
Normal Cover ◽  
Transitive Graphs

AbstractA graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s+1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s=1,2,3 or 4. Furthermore, if s=2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K5,5 minus a 1-factor, or K7,7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,2); if s=3, then X is a normal cover of the complete bipartite graph of order 4; if s=4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,3). As an application, we classify the tetravalent s-transitive graphs of order 2p2 for prime p.

scholarly journals On Finite Subnormal Cayley Graphs

10.37236/9934 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Shu Jiao Song
Keyword(s):  
Bipartite Graph ◽  
Cayley Graph ◽  
Complete Bipartite Graph ◽  
Cayley Graphs ◽  
Normal Cayley Graph ◽  
Normal Cover ◽  
Edge Transitive

In this paper we introduce and study a type of Cayley graph – subnormal Cayley graph. We prove that a subnormal 2-arc transitive Cayley graph is a normal Cayley graph or a normal cover of a complete bipartite graph $\mathbf{K}_{p^d,p^d}$ with $p$ prime. Then we obtain a generic method for constructing half-symmetric (namely edge transitive but not arc transitive) Cayley graphs.

scholarly journals Cores of Vertex Transitive Graphs

10.37236/3144 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
David E. Roberson
Keyword(s):  
Cayley Graphs ◽  
Transitive Graph ◽  
The Core ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs ◽  
Vertex Sets ◽  
Normal Cayley Graphs

A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets each of which induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions.

Sign in / Sign up

Export Citation Format

Share Document