Two sufficient conditions for non-normal Cayley graphs and their applications

2007 ◽  
Vol 50 (2) ◽  
pp. 201-216 ◽  
Author(s):  
Jin-xin Zhou ◽  
Yan-quan Feng
Keyword(s):  
Cayley Graphs ◽  
Sufficient Conditions ◽  
Normal Cayley Graphs

scholarly journals Locally Primitive Normal Cayley Graphs of Metacyclic Groups

10.37236/185 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jiangmin Pan
Keyword(s):  
Dihedral Group ◽  
Cayley Graphs ◽  
Complete Characterization ◽  
Metacyclic Group ◽  
Metacyclic Groups ◽  
Normal Cayley Graphs

A complete characterization of locally primitive normal Cayley graphs of metacyclic groups is given. Namely, let $\Gamma={\rm Cay}(G,S)$ be such a graph, where $G\cong{\Bbb Z}_m.{\Bbb Z}_n$ is a metacyclic group and $m=p_1^{r_1}p_2^{r_2}\cdots p_t^{r_t}$ such that $p_1 < p_2 < \dots < p_t$. It is proved that $G\cong D_{2m}$ is a dihedral group, and $val(\Gamma)=p$ is a prime such that $p|(p_1(p_1-1),p_2-1,\dots,p_t-1)$. Moreover, three types of graphs are constructed which exactly form the class of locally primitive normal Cayley graphs of metacyclic groups.

scholarly journals Two-distance transitive normal Cayley graphs

2021 ◽  
Author(s):  
Jun-Jie Huang ◽  
Yan-Quan Feng ◽  
Jin-Xin Zhou
Keyword(s):  
Cayley Graphs ◽  
Normal Cayley Graphs

scholarly journals Perfect State Transfer on Cayley Graphs over Dihedral Groups: The Non-Normal Case

10.37236/9184 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Xiwang Cao ◽  
Bocong Chen ◽  
San Ling
Keyword(s):  
Group Algebra ◽  
Quantum Information Processing ◽  
Cayley Graphs ◽  
Sufficient Conditions ◽  
Perfect State ◽  
Dihedral Groups ◽  
Normal Case ◽  
State Transfer ◽  
Perfect State Transfer ◽  
Necessary And Sufficient

Recently,  perfect state transfer (PST for short) on graphs has attracted great attention due to their applications in quantum information processing and quantum computations. Many constructions and results have been established through various graphs. However, most of the graphs previously investigated are abelian Cayley graphs. Necessary and sufficient conditions for Cayley graphs over dihedral groups having perfect state transfer were studied recently. The key idea in that paper is the assumption of the normality of the connection set. In those cases, viewed as an element in a group algebra, the connection set is in the center of the group algebra, which makes the situations just like in the abelian case. In this paper, we study the non-normal case. In this case, the discussion becomes more complicated. Using the representations of the dihedral group $D_n$,  we show that ${\rm Cay}(D_n,S)$ cannot have PST if $n$ is odd. For even integers $n$, it is proved that if ${\rm Cay}(D_n,S)$ has PST, then $S$ is normal.

Normal Cayley graphs of finite groups

1998 ◽  
Vol 41 (3) ◽  
pp. 242-251 ◽  
Author(s):  
Changqun Wang ◽  
Dianjun Wang ◽  
Mingyao Xu
Keyword(s):  
Finite Groups ◽  
Cayley Graphs ◽  
Normal Cayley Graphs

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.

Automorphisms of a Family of Cubic Graphs

2013 ◽  
Vol 20 (03) ◽  
pp. 495-506 ◽  
Author(s):  
Jin-Xin Zhou ◽  
Mohsen Ghasemi
Keyword(s):  
Automorphism Group ◽  
Positive Integer ◽  
Cayley Graph ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Cubic Graphs ◽  
Full Automorphism Group ◽  
Vertex Set ◽  
The Right ◽  
Normal Cayley Graphs

A Cayley graph Cay (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to [Formula: see text] for n=2, and to [Formula: see text] for n > 2. Furthermore, we determine all pairs of G and S such that Γn= Cay (G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.

Hamiltonian Cycles in Normal Cayley Graphs

2019 ◽  
Vol 35 (6) ◽  
pp. 1707-1714 ◽  
Author(s):  
Juan José Montellano-Ballesteros ◽  
Anahy Santiago Arguello
Keyword(s):  
Cayley Graphs ◽  
Hamiltonian Cycles ◽  
Normal Cayley Graphs

On Cayley graphs of completely 0-simple semigroups

2013 ◽  
Vol 11 (5) ◽  
Author(s):  
Shoufeng Wang ◽  
Yinghui Li
Keyword(s):  
Cayley Graphs ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

AbstractWe give necessary and sufficient conditions for various vertex-transitivity of Cayley graphs of the class of completely 0-simple semigroups and its several subclasses. Moreover, the question when the Cayley graphs of completely 0-simple semigroups are undirected is considered.

scholarly journals Tetravalent Non-Normal Cayley Graphs of Order $4p$

10.37236/207 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Full Automorphism Group ◽  
The Right ◽  
Normal Cayley Graphs

A Cayley graph ${\rm Cay}(G,S)$ on a group $G$ is said to be normal if the right regular representation $R(G)$ of $G$ is normal in the full automorphism group of ${\rm Cay}(G,S)$. In this paper, all connected tetravalent non-normal Cayley graphs of order $4p$ are constructed explicitly for each prime $p$. As a result, there are fifteen sporadic and eleven infinite families of tetravalent non-normal Cayley graphs of order $4p$.

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