Null graph

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, formula_1, is the unique graph having no vertices (hence its order is zero). It follows that formula_1 also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude formula_1 from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including formula_1 as a valid graph is useful depends on context. On the positive side, formula_1 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 formula_1 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 formula_1 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 "K"0). 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.
In category theory, the order-zero graph is, according to some definitions of "category of graphs," the initial object in the category. 
formula_1 does fulfill (vacuously) most of the same basic graph properties as does formula_9 (the graph with one vertex and no edges). As some examples, formula_1 is of size zero, it is equal to its complement graph formula_11, 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 formula_1.
Edgeless graph.
For each natural number "n", the edgeless graph (or empty graph) formula_13 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.
It is a 0-regular graph. The notation formula_13 arises from the fact that the "n"-vertex edgeless graph is the complement of the complete graph formula_15.