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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 ...

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 ...

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 ...

A graph is vertex transitive if $x \mapsto y$ by an automorphism. A graph is generously vertex transitive if $x \mapsto y \mapsto x$ by an automorphism. Simple facts: GVT $\rightarrow$ unimodular. ...

Consider an infinite graph that satisfies the following property: if any finite set of vertices is removed (and all the adjacent edges), then the resulting graph has only one infinite connected ...

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 ...