Singleton Graph is Unique

Theorem

The singleton graph $N_1$ is unique (up to isomorphism).

Proof

$N_1$ can be expressed as:

$N_1 := \struct {\set v, \O}$

where:

$\set v$ is the set of vertices

$\O$ is the set of edges, which is empty by definition.

Suppose there exists another singleton graph $N_1' = \struct {\set v', \O}$.

Let $\phi: \set v \to \set {v'}$ be the mapping from $N_1$ to $N_1'$ defined as:

$\map \phi v = v'$

From Mapping from Singleton is Injection, $\phi$ is an injection.

From Mapping to Singleton is Surjection, $\phi$ is an surjection.

Hence $\phi$ is a bijection by definition.

Finally it is noted that:

for each edge $\set {u, v} \in \map E {N_1}$, there exists an edge $\set {\map \phi u, \map \phi v} \in \map E {N_1'}$

holds vacuously

and:

for each edge $\set {u, v} \in \map E {N_1'}$, there exists an edge $\set {\map {\phi^{-1} } u, \map {\phi^{-1} } v} \in \map E {N_1}$

also holds vacuously.

Hence $N_1$ is isomorphic to $N_1'$ by definition.

Hence the result.

$\blacksquare$