Graphical Regular Representations of Non-Abelian Groups, I

1972 ◽  
Vol 24 (6) ◽  
pp. 993-1008 ◽  
Author(s):  
Lewis A. Nowitz ◽  
Mark E. Watkins
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Abelian Groups ◽  
Regular Representation ◽  
Abstract Group ◽  
Regular Representations ◽  
Graphical Regular Representation ◽  
Vertex Set

In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). IfXis such a graph with vertex setV(X)and automorphism groupA(X),we say thatXis agraphical regular representation(GRR) of a given abstract groupGif(I) G ≅ A(X) , and(II)A(X)acts onV(X) as a regular permutation group; that is, givenu, v∈V(X), there exists a uniqueφ∈A(X)for whichφ(u) =v.That for any abstract groupGthere exists a graphXsatisfying (I) is well-known (cf. [3]).

Graphical Regular Representations of Non-Abelian Groups, II

1972 ◽  
Vol 24 (6) ◽  
pp. 1009-1018 ◽  
Author(s):  
Lewis A. Nowitz ◽  
Mark E. Watkins
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Regular Representation ◽  
Affirmative Answer ◽  
Roman Numeral ◽  
Regular Representations ◽  
Graphical Regular Representation ◽  
Bold Face

The present paper is a sequel to the previous paper bearing the same title by the same authors [3] and which will be hereafter designated by the bold-face Roman numeral I. Further results are obtained in determining whether a given finite non-abelian group G has a graphical regular representation. In particular, an affirmative answer will be given if (|G|, 6) = 1.Inasmuch as much of the machinery of I will be required in the proofs to be presented and a perusal of I is strongly recommended to set the stage and provide motivation for this paper, an independent and redundant introduction will be omitted in the interest of economy.

scholarly journals Infinite Graphical Frobenius Representations

10.37236/7294 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Mark E. Watkins
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Polynomial Growth ◽  
Free Products ◽  
Locally Finite ◽  
Finitely Generated Groups ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Planar Maps ◽  
Transitive Graphs

A graphical Frobenius representation (GFR) of a Frobenius (permutation) group $G$ is a graph $\Gamma$ whose automorphism group Aut$(\Gamma)$ acts as a Frobenius permutation group on the vertex set of $\Gamma$, that is, Aut$(\Gamma)$ acts vertex-transitively with the property that all nonidentity automorphisms fix either exactly one or zero vertices and there are some of each kind. The set $K$ of all fixed-point-free automorphisms together with the identity is called the kernel of $G$. Whenever $G$ is finite, $K$ is a regular normal subgroup of $G$ (F. G. Frobenius, 1901), in which case $\Gamma$ is a Cayley graph of $K$. The same holds true for all the infinite instances presented here.Infinite, locally finite, vertex-transitive graphs can be classified with respect to (i) the cardinality of their set of ends and (ii) their growth rate. We construct families of infinite GFRs for all possible combinations of these two properties. There exist infinitely many GFRs with polynomial growth of degree $d$ for every positive integer $d$, and there exist infinite families of GFRs of exponential growth, both $1$-ended and infinitely-ended, that underlie infinite chiral planar maps. There also exist GFRs of free products of finitely many finitely generated groups. Graphs of connectivity 1 having a Frobenius automorphism group are characterized.

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 Reconstruction of homogeneous relational structures

2007 ◽  
Vol 72 (3) ◽  
pp. 792-802 ◽  
Author(s):  
Silvia Barbina ◽  
Dugald Macpherson
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Group Structure ◽  
Baire Category ◽  
Abstract Group ◽  
Relational Structures ◽  
Small Index Property ◽  
Categorical Structure ◽  
Small Index

This paper contains a result on the reconstruction of certain homogeneous transitive ω-categorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair (in a slightly non-standard sense, coming from Baire category).Reconstruction results give conditions under which the abstract group structure of the automorphism group Aut() of an ω-categorical structure determines the topology on Aut(), and hence determines up to bi-interpretability, by [1]; they can also give conditions under which the abstract group Aut() determines the permutation group ⟨Aut (), ⟩. so determines up to bi-definability. One such condition has been identified by M. Rubin in [12], and it is related to the definability, in Aut(), of point stabilisers. If the condition holds, the structure is said to have a weak ∀∃ interpretation, and Aut() determines up to bi-interpretability or, in some cases, up to bi-definability.A better-known approach to reconstruction is via the ‘small index property’: an ω-categorical stucture has the small index property if any subgroup of Aut() of index less than is open. This guarantees that the abstract group structure of Aut() determines the topology, so if is ω-categorical with Aut() ≅ Aut() then and are bi-interpretable.

scholarly journals On the asymptotic enumeration of Cayley graphs

2021 ◽  
Author(s):  
Joy Morris ◽  
Mariapia Moscatiello ◽  
Pablo Spiga
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Asymptotic Enumeration ◽  
Undirected Graphs ◽  
Regular Group ◽  
Cayley Digraph ◽  
Graphical Regular Representation ◽  
Proper Normal Subgroup

AbstractIn this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group R has a nontrivial proper normal subgroup N with the property that the automorphism group of the digraph fixes each N-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.

scholarly journals On the Existence of Frobenius Digraphical Representations

10.37236/7097 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Pablo Spiga
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Frobenius Group ◽  
Natural Extension ◽  
Partial Answer ◽  
Transitive Permutation Group ◽  
Frobenius Groups ◽  
Frobenius Complement ◽  
Vertex Set

A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A digraphical, respectively graphical, Frobenius representation, DFR and GFR for short, of a Frobenius group $F$ is a digraph, respectively graph, whose automorphism group as a group of permutations of the vertex set is $F$. The problem of classifying which Frobenius groups admit a DFR and GFR has been proposed by Mark Watkins and Thomas Tucker and is a natural extension of the problem of classifying which groups that have a digraphical, respectively graphical, regular representation.In this paper, we give a partial answer to a question of Mark Watkins and Thomas Tucker concerning Frobenius representations: "All but finitely many Frobenius groups with a given Frobenius complement have a DFR".  

scholarly journals A report on stable graphs

1973 ◽  
Vol 15 (2) ◽  
pp. 163-171 ◽  
Author(s):  
D. A. Holton
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Complete Graph ◽  
Symmetric Groups ◽  
Vertex Set

It is the aim of this paper to introduce a new concept relating various subgroups of the automorphism group of a graph to corresponding subgraphs. Throughout g will denote a (Michigan) graph on a vertex set V(¦V¦ =n) and Γ(g)=G will be the automorphism group of G considered as a permutation group on V.En, Cn, Dn and Sn are the identity, cyclic, dihedral, and symmetric groups acting on a set of size n, while Sp(q) is the permutation group of pq objects which is isomorphic to Sp but is q-fold in the sense that the objects are permuted q at a time [6]. H ≦ G means that H is a subgroup of G. Other group concepts can be found in Wielandt [7]. The graphs G1 ∪ G2, G1 + G2, G1 × G2, and G1[G2] along with their corresponding groups are as defined in, for example, Harary [4]. Finally we use Kn for the complete graph on n vertices.

On enhanced power graphs of finite groups

2018 ◽  
Vol 17 (08) ◽  
pp. 1850146 ◽  
Author(s):  
Sudip Bera ◽  
A. K. Bhuniya
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Cyclic Subgroups ◽  
Power Graph ◽  
Vertex Set ◽  
One To One

Given a group [Formula: see text], the enhanced power graph of [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are edge connected in [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text]. Here, we show that the graph [Formula: see text] is complete if and only if [Formula: see text] is cyclic; and [Formula: see text] is Eulerian if and only if [Formula: see text] is odd. We characterize all abelian groups and all non-abelian [Formula: see text]-groups [Formula: see text] such that [Formula: see text] is dominatable. Besides, we show that there is a one-to-one correspondence between the maximal cliques in [Formula: see text] and the maximal cyclic subgroups of [Formula: see text].

scholarly journals A SURVEY ON THE AUTOMORPHISM GROUPS OF THE COMMUTING GRAPHS AND POWER GRAPHS

2019 ◽  
pp. 729
Author(s):  
Mahsa Mirzargar
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
The Other ◽  
Power Graph ◽  
Commuting Graph ◽  
Vertex Set ◽  
Delta G ◽  
Nite Group

Let G be a nite group. The power graph P(G) of a group G is the graphwhose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

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