scholarly journals On finite dual Cayley graphs

2020 ◽  
Vol 18 (1) ◽  
pp. 595-602
Author(s):  
Jiangmin Pan
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Automorphism Groups ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Left Regular Representation ◽  
Left Regular ◽  
The Right

Abstract A Cayley graph \Gamma on a group G is called a dual Cayley graph on G if the left regular representation of G is a subgroup of the automorphism group of \Gamma (note that the right regular representation of G is always an automorphism group of \Gamma ). In this article, we study finite dual Cayley graphs regarding identification, construction, transitivity and such graphs with automorphism groups as small as possible. A few problems worth further research are also proposed.

scholarly journals Automorphism Groups of a Class of Cubic Cayley Graphs on Symmetric Groups

2017 ◽  
Vol 24 (04) ◽  
pp. 541-550
Author(s):  
Xueyi Huang ◽  
Qiongxiang Huang ◽  
Lu Lu
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Automorphism Groups ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Full Automorphism Group ◽  
Normal Cayley Graph ◽  
The Right

Let Sndenote the symmetric group of degree n with n ≥ 3, S = { cn= (1 2 ⋯ n), [Formula: see text], (1 2)} and Γn= Cay(Sn, S) be the Cayley graph on Snwith respect to S. In this paper, we show that Γn(n ≥ 13) is a normal Cayley graph, and that the full automorphism group of Γnis equal to Aut(Γn) = R(Sn) ⋊ 〈Inn(ϕ) ≅ Sn× ℤ2, where R(Sn) is the right regular representation of Sn, ϕ = (1 2)(3 n)(4 n−1)(5 n−2) ⋯ (∊ Sn), and Inn(ϕ) is the inner isomorphism of Sninduced by ϕ.

Automorphism Groups of Connected Cubic Cayley Graphs of Order 4p

2007 ◽  
Vol 14 (02) ◽  
pp. 351-359 ◽  
Author(s):  
Chuixiang Zhou ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Dihedral Group ◽  
Automorphism Groups ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Three Dimensional ◽  
Group A ◽  
The Right

For a prime p, let D4p be the dihedral group 〈a,b | a2p = b2 = 1, b-1ab = a-1〉 of order 4p, and Cay (G,S) a connected cubic Cayley graph of order 4p. In this paper, it is shown that the automorphism group Aut ( Cay (G,S)) of Cay (G,S) is the semiproduct R(G) ⋊ Aut (G,S), where R(G) is the right regular representation of G and Aut (G,S) = {α ∈ Aut (G) | Sα = S}, except either G = D4p (p ≥ 3), Sβ = {b,ab,apb} for some β ∈ Aut (D4p) and [Formula: see text], or Cay (G,S) is isomorphic to the three-dimensional hypercube Q3[Formula: see text] and G = ℤ4 × ℤ2 or D8.

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 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 A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A119

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2935
Author(s):  
Bo Ling ◽  
Wanting Li ◽  
Bengong Lou
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Automorphism Groups ◽  
Cayley Graphs ◽  
Vital Role ◽  
Alternating Group ◽  
Full Automorphism Group ◽  
Base Group

A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.

Cubic graphs admitting transitive non-abelian characteristically simple groups

2011 ◽  
Vol 54 (1) ◽  
pp. 113-123 ◽  
Author(s):  
Xiao-Hui Hua ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Cayley Graph ◽  
Regular Representation ◽  
Simple Groups ◽  
Symmetric Graph ◽  
Cubic Graphs ◽  
Vertex Transitive ◽  
Transitive Subgroup ◽  
The Right

AbstractLet Γ be a graph and let G be a vertex-transitive subgroup of the full automorphism group Aut(Γ) of Γ. The graph Γ is called G-normal if G is normal in Aut(Γ). In particular, a Cayley graph Cay(G, S) on a group G with respect to S is normal if the Cayley graph is R(G)-normal, where R(G) is the right regular representation of G. Let T be a non-abelian simple group and let G = Tℓ with ℓ ≥ 1. We prove that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-vertex-transitive cubic symmetric graph is G-normal. This result, among others, implies that a connected cubic symmetric Cayley graph on G is normal except for T ≅ A47 and a connected cubic G-symmetric graph is G-normal except for T ≅ A7, A15 or PSL(4, 2).

scholarly journals Elementary operators on prime C*-algebras II

1988 ◽  
Vol 30 (3) ◽  
pp. 275-284 ◽  
Author(s):  
Martin Mathieu
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Regular Representation ◽  
Bounded Operators ◽  
Weakly Compact ◽  
C Algebras ◽  
Left Regular Representation ◽  
Left Regular ◽  
Elementary Operators ◽  
The Right

Compact elementary operators acting on the algebra ℒ(H) of all bounded operators on some Hilbert space H were characterised by Fong and Sourour in [9]. Akemann and Wright investigated compact and weakly compact derivations on C*-algebras [1], and also studied compactness properties of the sum and the product of the right and the left regular representation of an element in a C*-algebra [2]. They used the concept of a compact Banach algebra element due to Vala [17]: an element a in a Banach algebra A is called compact if the mapping x → axa is compact on A. This notion has been further investigated by Ylinen [18, 19, 20], who showed in particular that a is a compact element of the C*-algebra A if x ↦ axa is weakly compact on A [19].

Representations of Compact Right Topological Groups

1993 ◽  
Vol 36 (3) ◽  
pp. 314-323 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Topological Group ◽  
Irreducible Representation ◽  
Regular Representation ◽  
List Type ◽  
Topological Groups ◽  
Regular Representations ◽  
Left Regular Representation ◽  
Left Regular ◽  
Enveloping Semigroups ◽  
The Right

AbstractCompact right topological groups arise naturally as the enveloping semigroups of distal flows. Recently, John Pym and the author established the existence of Haar measure μ on such groups, which invites the consideration of the regular representations. We start here by characterizing the continuous representations of a compact right topological group G, and are led to the conclusion that the right regular representation r is not continuous (unless G is topological). The domain of the left regular representation l is generally taken to be the topological centreor a tractable subgroup of it, furnished with a topology stronger than the relative topology from G (the goals being to have l both defined and continuous). An analysis of l and r on H = L2(G) for some non-topological compact right topological groups G shows, among other things, that: (i)for the simplest (perhaps) G generated by ℤ, (l, H) decomposes into one copy of each irreducible representation of ℤ and c copies of the regular representation.(ii)for the simplest (perhaps) G generated by the euclidean group of the plane , (l, H) decomposes into one copy of each of the continuous one-dimensional representations of and c copies of each continuous irreducible representation Ua,a > 0.(iii)when Λ(G) is not dense in G, it can seem very reasonable to regard r as a continuous representation of a related compact topological group, and also, G can be almost completely "lost" in the measure space (G, μ).

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

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.

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