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Approximation of the effective resistance on Cayley graph

Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...
Tomek Odrzygozdz's user avatar
16 votes
0 answers
362 views

Does every infinite, connected, locally finite, vertex-transitive graph have a leafless spanning tree?

My question is Let $G$ be an infinite, connected, locally finite, vertex-transitive graph. Must $G$ have the following substructures? i) a leafless spanning tree; ii) a spanning forest consisting ...
Agelos's user avatar
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7 votes
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220 views

Is there a Cayley graph with end space infinite and discrete?

A Cayley graph of a finitely generated group must be locally finite, and we know end spaces of locally finite graphs must be compact - so we can't have an infinite and discrete end space in this ...
violeta's user avatar
  • 407
3 votes
0 answers
393 views

What about a Cayley n-complex for n>2?

Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...
Sebastien Palcoux's user avatar
3 votes
0 answers
311 views

Induced graphs of Cayley graph

I have a Cayley graph $\mathrm{Cay}(G,S)$, its group presentation $G=\langle S | R \rangle$, and it becomes a metric graph by assigning a length equal to $1$ to each edge. I also have an induced ...
Miguel C.'s user avatar
0 votes
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90 views

Distances on spheres in Cayley graphs of non-amenable groups

Let $G$ be a non-amenable group (or perhaps more generally, a group with exponential growth). For any $\epsilon>0$, define the shell of radius r, $S_\epsilon(r)$, as the set of points that lie at a ...
user3521569's user avatar