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.

LOCALLY PRIMITIVE GRAPHS OF PRIME-POWER ORDER

2009 ◽  
Vol 86 (1) ◽  
pp. 111-122 ◽  
Author(s):  
CAI HENG LI ◽  
JIANGMIN PAN ◽  
LI MA
Keyword(s):  
Bipartite Graph ◽  
Cayley Graph ◽  
Complete Graph ◽  
Prime Power ◽  
Complete Bipartite Graph ◽  
Prime Power Order ◽  
Normal Cayley Graph ◽  
Normal Cover ◽  
Vertex Transitive

AbstractLet Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ×l, where Σ=Kpm with p prime or Σ is the Schläfli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph.

scholarly journals Normal Edge-Transitive Cayley Graphs of the Group U6n

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
A. Assari ◽  
F. Sheikhmiri
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
The Right ◽  
Edge Transitive

A Cayley graph of a group G is called normal edge-transitive if the normalizer of the right representation of the group in the automorphism of the Cayley graph acts transitively on the set of edges of the graph. In this paper, we determine all connected normal edge-transitive Cayley graphs of the group U6n.

On tetravalent s-regular Cayley graphs

2017 ◽  
Vol 16 (10) ◽  
pp. 1750195 ◽  
Author(s):  
Jing Jian Li ◽  
Bo Ling ◽  
Jicheng Ma
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Normal Cover

A Cayley graph [Formula: see text] is said to be core-free if [Formula: see text] is core-free in some [Formula: see text] for [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-regular if [Formula: see text] acts regularly on its [Formula: see text]-arcs. It is shown in this paper that if [Formula: see text], then there exist no core-free tetravalent [Formula: see text]-regular Cayley graphs; and for [Formula: see text], every tetravalent [Formula: see text]-regular Cayley graph is a normal cover of one of the three known core-free graphs. In particular, a characterization of tetravalent [Formula: see text]-regular Cayley graphs is given.

Frobenius REA-groups and their Cayley graphs

2019 ◽  
pp. 2150007
Author(s):  
Lei Wang ◽  
Shou Hong Qiao
Keyword(s):  
Cayley Graph ◽  
Automorphism Groups ◽  
Cayley Graphs ◽  
Frobenius Groups ◽  
Edge Transitive

In this paper, we determine the automorphism groups of a class of Frobenius groups, and then solve that under what condition they are REA-groups. As an application, we construct a type of normal edge-transitive Cayley graph.

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.

scholarly journals CLASSIFICATION OF TETRAVALENT -TRANSITIVE NONNORMAL CAYLEY GRAPHS OF FINITE SIMPLE GROUPS

2021 ◽  
pp. 1-9
Author(s):  
XIN GUI FANG ◽  
JIE WANG ◽  
SANMING ZHOU
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Cayley Graph ◽  
Finite Simple Group ◽  
Cayley Graphs ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Edge Transitive

Abstract A graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma $ and the set of s-arcs of $\Gamma $ , where for an integer $s \ge 1$ an s-arc of $\Gamma $ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma $ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$ . A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$ -arc-transitive but not $(\text{Aut} (\Gamma ), 3)$ -arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if $\Gamma $ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either $\Gamma $ is normal or G is one of the groups $\text{PSL}_2(11)$ , $\text{M} _{11}$ , $\text{M} _{23}$ and $A_{11}$ . However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

scholarly journals Cubic Edge-Transitive Bi-$p$-Metacirculants

10.37236/6417 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Yan-Li Qin ◽  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Group Of Automorphisms ◽  
Group A ◽  
Semisymmetric Graphs ◽  
Edge Transitive

A graph is said to be a bi-Cayley graph over a group $H$ if it admits $H$ as a group of automorphisms acting semiregularly on its vertices with two orbits. For a prime $p$, we call a bi-Cayley graph over a metacyclic $p$-group a bi-$p$-metacirculant. In this paper, the automorphism group of a connected cubic edge-transitive bi-$p$-metacirculant is characterized for an odd prime $p$, and the result reveals that a connected cubic edge-transitive bi-$p$-metacirculant exists only when $p=3$. Using this, a classification is given of connected cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic $3$-group. As a result, we construct the first known infinite family of cubic semisymmetric graphs of order twice a $3$-power.

scholarly journals Finite 1-Regular Cayley Graphs of Valency 5

2013 ◽  
Vol 2013 ◽  
pp. 1-3
Author(s):  
Jing Jian Li ◽  
Ben Gong Lou ◽  
Xiao Jun Zhang
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Normal Cover

Let and . We say is -regular Cayley graph if acts regularly on its arcs. is said to be core-free if is core-free in some . In this paper, we prove that if an -regular Cayley graph of valency is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free -regular Cayley graphs of valency .

scholarly journals Finite normal edge-transitive Cayley graphs

1999 ◽  
Vol 60 (2) ◽  
pp. 207-220 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Cayley Graph ◽  
Homomorphic Image ◽  
Quotient Group ◽  
Cayley Graphs ◽  
Sufficient Conditions ◽  
Nontrivial Group ◽  
Simple Quotient ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Edge Transitive

An approach to analysing the family of Cayley graphs for a finite group G is given which identifies normal edge-transitive Cayley graphs as a sub-family of central importance. These are the Cayley graphs for G for which a subgroup of automorphisms exists which both normalises G and acts transitively on edges. It is shown that, for a nontrivial group G, each normal edge-transitive Cayley graph for G has at least one homomorphic image which is a normal edge-transitive Cayley graph for a characteristically simple quotient group of G. Moreover, given a normal edge-transitive Cayley graph ΓH for a quotient group G/H, necessary and sufficient conditions are obtained for the existence of a normal edge-transitive Cayley graph Γ for G which has ΓH as a homomorphic image, and a method for obtaining all such graphs Γ is given.

scholarly journals Normal edge-transitive and \frac{1}{2}-arc-transitive cayley graphs on non-abelian groups of odd order 3pq, p and q are primes

2018 ◽  
Vol 49 (3) ◽  
pp. 183-194
Author(s):  
Ali Reza Ashrafi ◽  
Bijan Soleimani
Keyword(s):  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Prime Numbers ◽  
Graph Of Groups ◽  
Odd Order ◽  
Edge Transitive

Suppose $p$ and $q$ are odd prime numbers. In this paper, the connected Cayley graph of groups of order $3pq$, for primes $p$ and $q$, are investigated and all connected normal $\frac{1}{2}-$arc-transitive Cayley graphs of group of these orders will be classified.

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