Order 1 Simple Graph is Unique up to Isomorphism

Theorem
Let $G_1 = \struct {\map V {G_1}, \map E {G_1} }$ and $G_2 = \struct {\map V {G_2}, \map E {G_2} }$ be simple graphs of order $1$.

Then $G_1$ and $G_2$ are isomorphic.

Proof
There is only one bijection from $\map V {G_1}$ to $\map V {G_2}$.

There are no vertices adjacent to the sole vertex in $\map V {G_1}$

There are no vertices adjacent to the sole vertex in $\map V {G_2}$.

Hence the result, vacuously.