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I'm having a hard time seeing what $C_n$ would be for $n = 1$, or $n = 2$. Can someone clear up my confusion?

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2 Answers 2

$C_n$ is only defined for $n\geq 3$ because with $n=2$ or $n=1$ by definition you would have a tree without cycles.

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Thanks. Wikipedia does not mention this. –  Joel Sep 4 '12 at 22:06
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Why? Here's a uniform definition of $C_n$ for all $n$: it is the Cayley graph associated to the cyclic group $\mathbb{Z}/n\mathbb{Z}$ with generators $\{ 1, -1 \}$. This is a perfectly sensible definition even when $n = 1, 2$, it just gives a graph with multiple edges when $n = 2$ and a graph with a loop when $n = 1$ as in Paul's answer. –  Qiaochu Yuan Sep 5 '12 at 3:37

For $n\geq 3$, the cycle graph $C_n$ is the unique connected graph of order $n$ such that $\deg(u)=2$ for all vertices $u$. Note that $C_n$ is simple, that is, $C_n$ does not have loops or multiple edges.

So if you want to define $C_2$; if you want it to be a connected graph of order $2$ such that $\deg(u)=2$ for all vertices, then there will be multiple edges joining the two vertices of $C_2$. In particular, it is not simple. If you want $C_2$ to be connected and simple, then $C_2$ must be a tree. So $\deg(u)=1$ for all vertices $u$.

Similarly, for $C_1$, if you want it to be a connected graph of order $1$ such that $\deg(u)=2$ for all vertices, then there will be a loop at the single vertex of $C_1$. In particular, it is not simple.

So normally we only talk about $C_n$ when $n\geq 3$.

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