Is it true that for any integer $k\geq 3$ there are $\aleph_0$ many connected countably infinite, pairwise non-isomorphic $k$-regular graphs?
closed as off-topic by user44191, Ben Barber, Chris Godsil, Pace Nielsen, Dima Pasechnik Mar 10 at 9:36
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Take an $n$-cycle, add an infinite tree of the right degree at each vertex of the cycle (the vertex on the cycle having degree 2 less in the tree than the other vertices of the tree). This has only one cycle and it is of length $n$. So the graphs you get for two different values of $n$ are non-isomorphic.