Is it true that for any integer $k\geq 3$ there are $\aleph\_0$ many connected countably infinite, pairwise non-isomorphic $k$-regular graphs?



Infinite connected $k$-regular graphs
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.


