Questions tagged [cayley-graphs]
Cayley graphs are graphs obtained from a group $G$ in a such way that vertices are elements of the group and edges are added using some generating set $S\subseteq G$.
206
questions
2
votes
1
answer
27
views
acylindrically hyperbolic groups have hyperbolic Cayley graphs?
I am trying to make sense of Theorem 2.14 in Osin's article:
https://eta.impa.br/dl/071.pdf
It lists three equivalent conditions for a group $G$ to be acylindrically hyperbolic. Condition (c) reads:
(...
- 2,032
0
votes
1
answer
53
views
Are these Cayley graphs strongly connected?
My Background:
I have no formal training in graph theory. I'm just a group theory PhD student.
The Story:
I attended the New Perspectives in Computational Group Theory conference last week in honour ...
- 45.8k
1
vote
0
answers
22
views
Sub-multiplicativity of the growth function of a finetely generated group
I want to show that the growth function of a finetely generated group is Sub-multiplicative. In detail: Let $G$ be a finite and $S$ a finite generating set.
I want to proof that $\forall r,r'\in\...
- 31
0
votes
0
answers
51
views
Is there some higher-dimensional polytope that represents the Rubik's Cube group?
I recently found a Pocket Cube and while trying to find instructions on how to solve it, I got sucked into the whole Rubik maths rabbit hole. I was wondering if the Rubik's group can be represented by ...
- 101
0
votes
1
answer
60
views
How would a category theorist describe the Cayley graph of a group w.r.t. a subset?
Background:
The question at hand is in line with previous questions of mine, such as:
How would a category theorist describe Green's relations?
Describing the Wreath product categorically.
I ask ...
- 45.8k
0
votes
0
answers
29
views
Cayley Graph of permutations
Let $[p]=\{1,..,p\}$ where $p\in \mathbb{N}$. Let $P(n,r)$ denote the set of all injective functions from $[r]$ to $[n]$ and write a typical element as $\sigma=[\sigma(1),...,\sigma(r)]$ where $1\leq\...
1
vote
0
answers
124
views
Growh generating function for finite Heisenberg groups
Take standard 2d-Heisenberg group over finite ring Z/p. Choose standard generators $x_i, y_i$.
Consider generating polynomial for growth: $ g(t) = \sum_i g_i t^i $ , where $g_i$ are the ball sizes.
...
2
votes
1
answer
65
views
Schutzenberger graphs of an Inverse Semigroup?
I recently came across the idea of extending the well-known Cayley graph construction for semigroups and learned that the outcome does not have all the expected properties even for the nice classes of ...
- 18.4k
1
vote
1
answer
52
views
Spectrum of circulant block matrix of circulant blocks (Adjacency matrix of discrete torus)
I am currently investigating the spectrum of a matrix $M \in \mathbb{R}^{12 \times 12}$. The matrix has the following form,
$$
M = \begin{bmatrix}
0 & 1 & 0 & 1 & 1 & 0 &...
- 139
0
votes
0
answers
35
views
How Cayley diagram change without requirement that actions are determined?
The question is from the book "Visual group theory" by Nathan Carter. The book doesn't give an answer to this exercise so I post it here. It is the Exercise 2.14 at the page 24 of the pdf ...
- 101
3
votes
1
answer
133
views
Diameter of the Cayley graph of $\mathbb{Z}_n$
So, I have $G = \mathbb{Z}_n$ and $T_n = \{g\in\mathbb{Z}_n: \operatorname{gcd}(g,n) = 1\} = \mathbb{Z}_n^*$.
I need to find the diameter of $\Gamma(G,T_n)$.
What have I tried: we can obviously try to ...
3
votes
1
answer
66
views
Questions about harmonic functions on Cayley graphs
I am reading A proof of Gromov’s theorem by Terence Tao, where I encountered harmonic (and Lipschitz) functions on Cayley graphs, here is the definition:
Let $G$ be an infinite group generate by a ...
- 328
0
votes
0
answers
34
views
Closest Equivalent to Cayley Graphs for Partial Groupoids?
[A partial groupoid (half-magma) is a set S equipped with a (single-valued) partial binary operation, as in Bruck's Survey of Binary Systems.]
This question may be nonsensical, given that the duality ...
- 31
0
votes
0
answers
93
views
Cayley tree reachable nodes proof
Imagine being in the central node of a Cayley Tree graph like this one: Cayle Tree, K =3 and D = 5
For a number of reachable t steps (t<D) prove that the reacheable nodes from the center equal:
$$ ...
2
votes
0
answers
134
views
Cayley graph of a finitely generated group $G$
Good time of day!
I'm trying to show that: "The Cayley graph of a finitely generated group $G$ is quasi-isometric to a line if and only if $G$ has a cyclic subgroup of finite index".
My ...
- 411