In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph").

Order-zero graph

The order-zero graph, K₀, is the unique graph having no vertices (hence its order is zero). It follows that K₀ also has no edges. Thus the null graph is a regular graph of degree zero. 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). On the negative side, including K₀ as a graph requires that many well-defined formulas for graph properties include exceptions for it (for example, either "counting all strongly connected components of a graph" becomes "counting all non-null strongly connected components of a graph", or the definition of connected graphs has to be modified not to include K₀). To avoid the need for such exceptions, it is often assumed in literature that the term graph implies "graph with at least one vertex" unless context suggests otherwise.^[1][2]

In category theory, the order-zero graph is, according to some definitions of "category of graphs," the initial object in the category.

K₀ does fulfill (vacuously) most of the same basic graph properties as does K₁ (the graph with one vertex and no edges). As some examples, K₀ is of size zero, it is equal to its complement graph K₀, a forest, and a planar graph. It may be considered undirected, directed, or even both; when considered as directed, it is a directed acyclic graph. And it is both a complete graph and an edgeless graph. However, definitions for each of these graph properties will vary depending on whether context allows for K₀.

Edgeless graph

For each natural number n, the edgeless graph (or empty graph) K_n of order n is the graph with n vertices and zero edges. An edgeless graph is occasionally referred to as a null graph in contexts where the order-zero graph is not permitted.^[1][2]

It is a 0-regular graph. The notation K_n arises from the fact that the n-vertex edgeless graph is the complement of the complete graph K_n.