Some authors exclude K0 from consideration as a graph (either by definition, or more simply as a matter of convenience).
Whether including K0 as a valid graph is useful depends on context.
On the positive side, K0 follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair (V, E) for which the vertex and edge sets, V and E, are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures K0 is useful for defining the base case for recursion (by treating the null tree as the child of missing edges in any non-null binary tree, every non-null binary tree has exactly two children).
However, definitions for each of these graph properties will vary depending on whether context allows for K0.
The notation Kn arises from the fact that the n-vertex edgeless graph is the complement of the complete graph Kn.