All Questions
16 questions
1
vote
0
answers
340
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Random walk on non-abelian free group
Let $F_2$ be the free non-abelian group with generators $a, b\in F_2$.
Has the "random walk" where we start with the identity and then multiply it by $a$ or $b$ or $a^{-1}$ or $b^{-1}$ ...
- 63.9k
2
votes
1
answer
181
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Generators of sandpile groups of wheel graphs
In the paper "On the Sandpile Group of a Graph" by Cori and Rossin one can find a result related to the structure of the sandpile group of $W_n$. Is there a way to provide a set of ...
- 298
1
vote
0
answers
126
views
Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
- 63.9k
7
votes
2
answers
415
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Graph which do not satisfy a pseudo-Poincaré inequality
Say that an infinite (connected) graph (with vertices of bounded degree) satisfies a $\ell_1$-pseudo-Poincaré inequality if there is a constant $C>0$ so that for any $n \in \mathbb{N}$ for any ...
- 28k
4
votes
0
answers
236
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Groups inducing edge-colorings on graphs. Is this concept known?
Are the following concepts known in graph/group theory, and if Yes, what are they called and where to read about them? Because I do not know better, I gave them placeholder names for now.
1. ...
- 13.6k
3
votes
0
answers
231
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What is known about "graph algebras"?
In lack for a better name I call a "graph algebra" a simple undirected graph $G=(V,E)$ and a binary mapping $+:E \rightarrow V$ such that:
(1) For all edges $(a,b)$ we have: $a+b \in N(a) \cap N(b)$, ...
CommunityBot
- 1
10
votes
1
answer
269
views
Edge-transitive Cayley graphs of $S_n$
I came across the following question which I haven't seen before:
Question. Fix $k\ge 3$. For infinitely many $n$, does there exists a generating set $\langle R_n \rangle = S_n$, $|R_n|=k$, such ...
- 17k
5
votes
0
answers
169
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In the literature on infinite graphs, are there results on "periodizable" graphs?
Let $G=(V,E)$ be a connected countably infinite $k$-regular simple graph (no loops or multiple edges). For $A$ a finite subset of $V$, let me denote by $G_A=(A,E_A)$ the induced subgraph with vertex ...
6
votes
1
answer
899
views
Reconstruction Conjecture: Group theoretic formulation
As we read from wiki, informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs.
Is there a group-theoretic formulation of this conjecture?
...
- 3,686
2
votes
0
answers
187
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Classification of Automorphism set of a Regular graph
Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation matrices)....
CommunityBot
- 1
5
votes
2
answers
439
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Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$
If $\Gamma = C(G, S)$ is the (undirected) Cayley graph of a finite group $G$ with generating set $S$, then $G \le \operatorname{Aut}(\Gamma)$, the "full" automorphism group of $\Gamma$.
When is it ...
- 406
5
votes
1
answer
537
views
Which hyperbolic tilings are Cayley graphs?
I realise the question is easy but after asking to a few people (and never getting a clear answer), I thought it could be instructive to ask it here:
Given a regular tiling of the hyperbolic plane is ...
- 1,033
3
votes
2
answers
1k
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A structure of the group of automorphisms of an infinite binary tree
My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary ...
- 38.6k
1
vote
0
answers
611
views
Is the automorphism group of a homogeneous (locally finite) tree unimodular?
I have seen somewhere (that I don't remember now) that the (full) automorphism group of a k-regular tree is unimodular. I assume a k-regular tree is the same thing as the homogeneous tree of degree k (...
CommunityBot
- 1
3
votes
0
answers
135
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Groups acting on non-locally-finite trees with independence and specified local actions
Suppose I have a biregular tree $T_{m, n}$ (not necessarily locally finite), with distinct cardinal numbers $m, n$, so Aut$(T_{m, n})$ acts on $T_{m, n}$ without inversion. Let $V_m$ be those vertices ...
- 313
12
votes
3
answers
552
views
Estimate on currents in Cayley graphs
Take a Cayley graph $\Gamma$ (thought of as an electrical network with all edges having equal resistance) and break one edge $e$ and put a battery there. (Assume the graph has only one end* so that ...
- 4,432