Graph is 0-Regular iff Edgeless

Theorem

Let $G$ be a graph.

Then $G$ is $0$-regular graph if and only if $G$ is edgeless.

Proof

Sufficient Condition

Let $G$ be an edgeless graph.

By definition of edgeless graph, $G$ has no edges.

That means every vertex of $G$ is incident with no edges.

That is, every vertex of $G$ is of degree $0$.

Hence the result, by definition of $0$-regular graph.

$\Box$

Necessary Condition

Let $G$ be a $0$-regular graph.

Then every vertex is incident with no edges.

So as there are no vertices with incident edges, there can be no edges.

Hence $G$ is an edgeless graph.

$\blacksquare$