Cycle Graph of Order 3 is Complete Graph

Theorem

Let $C_3$ denote the cycle graph of order $2$.

Then $C_3$ is the complete graph of of order $3$.

Proof

Let the vertex set of $C_3$ is $\set {v_1, v_2, v_3}$.

By definition of cycle graph, $C_3$ consists of the cycle $v_1 v_2 v_3 v_1$.

It is seen by inspection that:

$v_1$ is adjacent to $v_2$ and $v_3$

$v_2$ is adjacent to $v_1$ and $v_3$

$v_3$ is adjacent to $v_1$ and $v_2$.

Hence the result by definition of complete graph.

$\blacksquare$