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How many infinite connected graphs are there, that satisfy:
-Every node has $n$ neighbours
-Every node is symmetric, ie take two graphs and any one point on each, then we can strecth one graph to have the two points ontop of eachother, aswell as having all other nodes and vertices coincide (is there a name for this?)

For instance for n=2, only an infinite line with countably infinite nodes on it is possible

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    ... or in general, hyperbolic tessellations of $k$-gons (k>6, say) meeting in $n$s for a countably infinite familiy of vertex-transitive $n$-regular graphs for each $n\ge 3$.2011-11-08

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