scholarly journals Some New Groups which are not CI-groups with Respect to Graphs

10.37236/6541 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Ted Dobson
Keyword(s):  
Dihedral Group ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Group Automorphism ◽  
Nonabelian Group

A group $G$ is a CI-group with respect to graphs if two Cayley graphs of $G$ are isomorphic if and only if they are isomorphic by a group automorphism of $G$.  We show that an infinite family of groups which include $D_n\times F_{3p}$ are not CI-groups with respect to graphs, where $p$ is prime, $n\not = 10$ is relatively prime to $3p$, $D_n$ is the dihedral group of order $n$, and $F_{3p}$ is the nonabelian group of order $3p$.

scholarly journals Semidirect product groups with Abelian automorphism groups

1987 ◽  
Vol 42 (1) ◽  
pp. 84-91 ◽  
Author(s):  
M. J. Curran
Keyword(s):  
Automorphism Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Semidirect Product ◽  
Dihedral Group ◽  
Automorphism Groups ◽  
Infinite Family ◽  
Central Automorphism ◽  
Nonabelian Group ◽  
Semidirect Product Groups

AbstractMiller's group of order 64 is a smallest example of a nonabelian group with an abelian automorphism group, and is the first in an infinite family of such groups formed by taking the semidirect product of a cyclic group of order 2m (m ≥ 3) with a dihedral group of order 8. This paper gives a method for constructing further examples of non abelian 2-groups which have abelian automorphism groups. Such a 2-group is the semidirect product of a cyclic group and a special 2-group (satisfying certain conditions). The automorphism group of this semidirect product is shown to be isomorphic to the central automorphism group of the corresponding direct product. The conditions satisfied by the special 2-group are determined by establishing when this direct product has an abelian central automorphism group.

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

Integral Cayley graphs over a certain nonabelian group

2021 ◽  
pp. 1-12
Author(s):  
Tao Cheng ◽  
Lihua Feng ◽  
Weijun Liu
Keyword(s):  
Cayley Graphs ◽  
Nonabelian Group

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.

scholarly journals RAMANUJAN CAYLEY GRAPHS OF FROBENIUS GROUPS

2016 ◽  
Vol 94 (3) ◽  
pp. 373-383 ◽  
Author(s):  
MIKI HIRANO ◽  
KOHEI KATATA ◽  
YOSHINORI YAMASAKI
Keyword(s):  
Dihedral Group ◽  
Cayley Graphs ◽  
The Other ◽  
Frobenius Groups ◽  
Quadratic Polynomials ◽  
Normal Case ◽  
Ramanujan Graphs ◽  
Formula One ◽  
The Family

We determine a bound for the valency in a family of dihedrants of twice odd prime orders which guarantees that the Cayley graphs are Ramanujan graphs. We take two families of Cayley graphs with the underlying dihedral group of order $2p$: one is the family of all Cayley graphs and the other is the family of normal ones. In the normal case, which is easier, we discuss the problem for a wider class of groups, the Frobenius groups. The result for the family of all Cayley graphs is similar to that for circulants: the prime $p$ is ‘exceptional’ if and only if it is represented by one of six specific quadratic polynomials.

scholarly journals Groups all of whose undirected Cayley graphs are determined by their spectra

2016 ◽  
Vol 15 (09) ◽  
pp. 1650175 ◽  
Author(s):  
Alireza Abdollahi ◽  
Shahrooz Janbaz ◽  
Mojtaba Jazaeri
Keyword(s):  
Adjacency Matrix ◽  
Sylow Subgroup ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Solvable Groups ◽  
Adjacency Spectrum

The adjacency spectrum [Formula: see text] of a graph [Formula: see text] is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph [Formula: see text] is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group [Formula: see text] is Cay-DS if every two cospectral Cayley graphs of [Formula: see text] are isomorphic. In this paper, we study finite DS groups and finite Cay-DS groups. In particular we prove that a finite DS group is solvable, and every non-cyclic Sylow subgroup of a finite DS group is of order [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. We also give several infinite families of non-Cay-DS solvable groups. In particular we prove that there exist two cospectral non-isomorphic [Formula: see text]-regular Cayley graphs on the dihedral group of order [Formula: see text] for any prime [Formula: see text].

scholarly journals Bounds for Domination Parameters in Cayley Graphs on Dihedral Group

2012 ◽  
Vol 02 (01) ◽  
pp. 5-10 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
G. Kalaimurugan
Keyword(s):  
Dihedral Group ◽  
Cayley Graphs

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.

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