Heptavalent Symmetric Graphs of Order 16p

2017 ◽  
Vol 24 (03) ◽  
pp. 453-466 ◽  
Author(s):  
Songtao Guo ◽  
Hailong Hou ◽  
Yong Xu
Keyword(s):  
Automorphism Group ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Symmetric Graphs ◽  
Normal Cayley Graphs

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. We classify connected heptavalent symmetric graphs of order 16p for each prime p. As a result, there are two such sporadic graphs with p = 3 and 7, and an infinite family of 1-regular normal Cayley graphs on the group [Formula: see text] with 7|(p – 1).

A 2-ARC TRANSITIVE PENTAVALENT CAYLEY GRAPH OF 

2016 ◽  
Vol 93 (3) ◽  
pp. 441-446 ◽  
Author(s):  
BO LING ◽  
BEN GONG LOU
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Discrete Math ◽  
Simple Groups ◽  
Alternating Group ◽  
Full Automorphism Group ◽  
Symmetric Graphs

Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.

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.

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 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 $4p$.

scholarly journals Cubic Vertex-Transitive Non-Cayley Graphs of Order $8p$

10.37236/2087 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Jin-Xin Zhou ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Infinite Family ◽  
The Other ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Unique Graph ◽  
Vertex Transitive Graph

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertex-transitive non-Cayley graphs of order $8p$ are classified for each prime $p$. It follows from this classification that there are two sporadic and two infinite families of such graphs, of which the sporadic ones have order $56$,  one infinite family exists for every prime $p>3$ and the other family exists if and only if $p\equiv 1\mod 4$. For each family there is a unique graph for a given order.

scholarly journals Cubic arc-transitive Cayley graphs on Frobenius groups

2018 ◽  
Vol 17 (07) ◽  
pp. 1850126 ◽  
Author(s):  
Hailin Liu ◽  
Lei Wang
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Frobenius Groups

A Cayley graph [Formula: see text] is called arc-transitive if its automorphism group [Formula: see text] is transitive on the set of arcs in [Formula: see text]. In this paper, we give a characterization of cubic arc-transitive Cayley graphs on a class of Frobenius groups.

scholarly journals A Family of Half-Transitive Graphs

10.37236/3140 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Jing Chen ◽  
Cai Heng Li ◽  
Ákos Seress
Keyword(s):  
Cayley Graphs ◽  
Infinite Family ◽  
Transitive Graphs

We construct an infinite family of half-transitive graphs, which contains infinitely many Cayley graphs, and infinitely many non-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 An infinite family of cubic nonnormal Cayley graphs on nonabelian simple groups

2018 ◽  
Vol 341 (5) ◽  
pp. 1282-1293
Author(s):  
Jiyong Chen ◽  
Binzhou Xia ◽  
Jin-Xin Zhou
Keyword(s):  
Cayley Graphs ◽  
Infinite Family ◽  
Simple Groups

SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS NEED NOT BE CAYLEY GRAPHS

2001 ◽  
Vol 33 (6) ◽  
pp. 653-661 ◽  
Author(s):  
CAI HENG LI ◽  
CHERYL E. PRAEGER
Keyword(s):  
Wreath Product ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Permutation Groups ◽  
General Study ◽  
Product Action ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such graphs. The nature of the construction was suggested by a general study of the structure of self-complementary, vertex-transitive graphs. It involves the product action of a wreath product of permutation groups.

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