Cycle Graph of Order 1 is Loop-Graph
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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.
$\blacksquare$