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If I have a single vertex with a self-loop. Do we call that a circuit? Because we "loop" around itself once?

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A cycle in a graph is, according to Wikipedia,

An edge set that has even degree at every vertex; also called an even edge set or, when taken together with its vertices, an even subgraph.

In your case, the single vertex has a degree of 2, which is even. Therefore the self-loop is a cycle in your graph.

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Note that, generally, "cycle" and "circuit" have different meanings. –  Douglas S. Stones Apr 13 '13 at 12:47

I'd lean towards calling it a circuit. However, when it comes to these technical and isolated cases, it's often best to simply choose whichever definition makes your work easiest to read.

The distinction between defining a loop as a circuit or not should be a very minor consideration in a larger body of work. The larger body of work is what's important. As long as we state our definition unambiguously and are consistent throughout, it's not a real problem.

To illustrate, if you go through my papers and look at the different ways I've defined a Latin square, you'll see that the definitions are inconsistent. I just pick whichever definition makes the paper easiest to read. I could use up the readers time in being consistent, but instead I just get to the real point of the paper.

However, in some cases these slight distinctions can matter. For example, Counting loopy graphs with given degrees by McKay and Greenhill, the distinction between loops contributing 1 or 2 to the degree of a vertex made a non-negligible difference in the case of asymptotic enumeration of graphs with a given degree sequence. The author's responded to this situation by analysing both cases.

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