scholarly journals On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

scholarly journals ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

2004 ◽  
Vol 14 (05n06) ◽  
pp. 677-702 ◽  
Author(s):  
V. S. GUBA
Keyword(s):  
Growth Rate ◽  
Lower Bound ◽  
Cayley Graph ◽  
Upper Bound ◽  
Growth Function ◽  
Vertex Degree ◽  
General Growth ◽  
Thompson’S Group ◽  
Average Vertex ◽  
Thompson's Group F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

scholarly journals Random spanning trees of Cayley graphs and an associated compactification of semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 611-620
Author(s):  
Steven N. Evans
Keyword(s):  
Random Walk ◽  
Cayley Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Finite Tree ◽  
Generating Set ◽  
Countably Infinite

A sequential construction of a random spanning tree for the Cayley graph of a finitely generated, countably infinite subsemigroup V of a group G is considered. At stage n, the spanning tree T isapproximated by a finite tree Tn rooted at the identity.The approximation Tn+1 is obtained by connecting edges to the points of V that are not already vertices of Tn but can be obtained from vertices of Tn via multiplication by a random walk step taking values in the generating set of V. This construction leads to a compactification of the semigroup V inwhich a sequence of elements of V that is not eventually constant is convergent if the random geodesic through the spanning tree T that joins the identity to the nth element of the sequence converges in distribution as n→∞. The compactification is identified in a number of examples. Also, it is shown that if h(Tn) and #(Tn) denote, respectively, the height and size of the approximating tree Tn, then there are constants 0<ch≤1 and 0≥c# ≤log2 such that limn→∞ n–1 h(Tn)= ch and limn→∞n–1 log# (Tn)= c# almost surely.

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

scholarly journals Dominating sets in Cayley graphs on $ Z_{n} $

2007 ◽  
Vol 38 (4) ◽  
pp. 341-345 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
I. Rani
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Generating Set ◽  
Minimal Dominating Set

A Cayley graph is a graph constructed out of a group $ \Gamma $ and its generating set $ A $. In this paper we attempt to find dominating sets in Cayley graphs constructed out of $ Z_{n} $. Actually we find the value of domination number for $ Cay(Z_{n}, A) $ and a minimal dominating set when $ |A| $ is even and further we have proved that $ Cay(Z_{n}, A) $ is excellent. We have also shown that $ Cay(Z_{n}, A) $ is $ 2- $excellent, when $ n = t(|A|+1)+1 $ for some integer $ t, t>0 $.

scholarly journals Cayley Graphs of Order 27p Are Hamiltonian

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Ebrahim Ghaderpour ◽  
Dave Witte Morris
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Hamiltonian Cycle ◽  
Cayley Graphs ◽  
Generating Set ◽  
A Finite Group

Suppose that G is a finite group, such that |G|=27p, where p is prime. We show that if S is any generating set of G, then there is a Hamiltonian cycle in the corresponding Cayley graph Cay (G;S).

The girth of Cayley graphs over Sylow 2-subgroups of the symmetric groups S2n with diagonal bases

2019 ◽  
Vol 18 (12) ◽  
pp. 1950237
Author(s):  
Bartłomiej Pawlik
Keyword(s):  
Symmetric Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Polynomial Representation ◽  
Generating Set ◽  
Minimal Generating Set

A diagonal base of a Sylow 2-subgroup [Formula: see text] of symmetric group [Formula: see text] is a minimal generating set of this subgroup consisting of elements with only one nonzero coordinate in the polynomial representation. For different diagonal bases, Cayley graphs over [Formula: see text] may have different girths (i.e. minimal lengths of cycles). In this paper, all possible values of girths of Cayley graphs over [Formula: see text] with diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented.

scholarly journals FOREST DIAGRAMS FOR ELEMENTS OF THOMPSON'S GROUP F

2005 ◽  
Vol 15 (05n06) ◽  
pp. 815-850 ◽  
Author(s):  
JAMES M. BELK ◽  
KENNETH S. BROWN
Keyword(s):  
Cayley Graph ◽  
Real Line ◽  
Unit Interval ◽  
Minimum Length ◽  
Generating Set ◽  
The Real ◽  
Positive Elements ◽  
Thompson’S Group ◽  
Standard Action ◽  
Thompson's Group F

We introduce forest diagrams to represent elements of Thompson's group F. These diagrams relate to a certain action of F on the real line in the same way that tree diagrams relate to the standard action of F on the unit interval. Using forest diagrams, we give a conceptually simple length formula for elements of F with respect to the {x0,x1} generating set, and we discuss the construction of minimum-length words for positive elements. Finally, we use forest diagrams and the length formula to examine the structure of the Cayley graph of F.

scholarly journals Automorphisms and isomorphisms of enhanced hypercubes

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2805-2812
Author(s):  
Lu Lu ◽  
Qiongxiang Huang
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Standard Basis ◽  
Generating Set ◽  
Folded Hypercube

Let Zn2 be the elementary abelian 2-group, which can be viewed as the vector space of dimension n over F2. Let {e1,..., en} be the standard basis of Zn2 and ?k = ek +...+ en for some 1 ? k ? n-1. Denote by ?n,k the Cayley graph over Zn2 with generating set Sk = {e1,..., en,?k}, that is, ?n,k = Cay(Zn2,Sk). In this paper, we characterize the automorphism group of ?n,k for 1 ? k ? n-1 and determine all Cayley graphs over Zn2 isomorphic to ?n,k. Furthermore, we prove that for any Cayley graph ? = Cay(Zn2,T), if ? and ?n,k share the same spectrum, then ? ? ?n,k. Note that ?n,1 is known as the so called n-dimensional folded hypercube FQn, and ?n,k is known as the n-dimensional enhanced hypercube Qn,k.

scholarly journals Metric dimensions of minor excluded graphs and minor exclusion in groups

2015 ◽  
Vol 25 (04) ◽  
pp. 541-554 ◽  
Author(s):  
Mikhail I. Ostrovskii ◽  
David Rosenthal
Keyword(s):  
Cayley Graph ◽  
Finite Graph ◽  
Infinite Graph ◽  
Group Property ◽  
Free Products ◽  
Generating Set ◽  
Finitely Generated Groups ◽  
Metric Dimensions ◽  
A Minor ◽  
Nagata Dimension

An infinite graph Γ is minor excluded if there is a finite graph that is not a minor of Γ. We prove that minor excluded graphs have finite Assouad–Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators, like ℤ3; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.

scholarly journals The Energy of Cayley Graphs for Symmetric Groups of Order 24

2020 ◽  
pp. 1-6
Author(s):  
Amira Fadina Ahmad Fadzil ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Finite Graph ◽  
Energy Of Graph ◽  
A Finite Group

A Cayley graph of a finite group G with respect to a subset S of G is a graph where the vertices of the graph are the elements of the group and two distinct vertices x and y are adjacent to each other if xy−1 is in the subset S. The subset of the Cayley graph is inverse closed and does not include the identity of the group. For a simple finite graph, the energy of a graph can be determined by summing up the positive values of the eigenvalues of the adjacency matrix of the graph. In this paper, the graph being studied is the Cayley graph of symmetric group of order 24 where S is the subset of S4 of valency up to two. From the Cayley graphs, the eigenvalues are calculated by constructing the adjacency matrix of the graphs and by using some properties of special graphs. Finally, the energy of the respected Cayley graphs is computed and presented. Keywords: energy of graph; cayley graph; symmetric groups

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