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Are infinite cycle graphs just "straight lines"?

I mean are they of the form: $\cdots \bullet - \bullet - \bullet - \bullet - \bullet - \cdots $

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    Unlike infinite cyclic groups, I'd expect any "cyclic graph" to have, you know, a cycle somewhere. I suppose there is a sense in which the line graph is like a finite cyclic graph, in that every edge has two neighbors and it is connected, but It really depends on the *dedinition* of cyclic graph.2012-12-25
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    It is hard to say if $\mathbb Z \cup \infty$ is an infinite cycle graph-which elements does $\infty$ connect to? What is your definition of a cycle graph? Given the definition, the answer might be available.2012-12-25

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