Cycle Graph of Order 1 is Loop-Graph

Theorem
Let $C_1$ denote the cycle graph of order $1$.

Then $C_1$ is a loop-graph.

Proof
By definition, the vertex set of $C_1$ is singleton, $\set v$, say.

The only vertex of $C_1$ that an edge can be incident to is $v$.

Hence there exists an edge which is incident to $v$ at both ends.

That is, $C_1$ has a loop.

Hence the result by definition of loop-graph.