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

Order-zero graph (null graph)
Vertices	0
Edges	0
Radius	0
Diameter	0
Girth	$\infty$
Automorphisms	1
Chromatic number	0
Chromatic index	0
Genus	0
Properties	Integral Symmetric
Notation	$K_0$

The order-zero graph, $K_0$, is the unique graph having no vertices (hence its order is zero). It follows that $K_0$ also has no edges. Some authors exclude $K_0$ from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including $K_0$ as a valid graph is useful depends on context. On the positive side, $K_0$ 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 $K_0$ 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 $K_0$ 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.

$K_0$ does fulfill (vacuously) most of the same basic graph properties as does $K_1$ (the graph with one vertex and no edges). As some examples, $K_0$ is of size zero, it is equal to its complement graph $\overline{K_0}$, 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 $K_0$.

Edgeless graph

Edgeless graph (empty graph, null graph)
Vertices	n
Edges	0
Radius	0
Diameter	0
Girth	$\infty$
Automorphisms	n!
Chromatic number	1
Chromatic index	0
Genus	0
Properties	Integral Symmetric
Notation	$\overline K_n$

For each natural number n, the edgeless graph (or empty graph) $\overline K_n$ 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]

The notation $\overline K_n$ arises from the fact that the n-vertex edgeless graph is the complement of the complete graph $K_n$.

See also

• Glossary of graph theory
• Cycle graph
• Path graph

Notes

1. Weisstein, Eric W., "Empty Graph", MathWorld.
2. Weisstein, Eric W., "Null Graph", MathWorld.

References

• Harary, F. and Read, R. (1973), "Is the null graph a pointless concept?", Graphs and Combinatorics (Conference, George Washington University), Springer-Verlag, New York, NY.