Empty graph

Order-zero graph (null graph)
Vertices	0
Edges	0
Girth	${\displaystyle \infty }$
Automorphisms	1
Chromatic number	0
Chromatic index	0
Genus	0
Properties	Integral Symmetric
Notation	${\displaystyle K_{0}}$

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

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").

The order-zero graph, ${\displaystyle K_{0}}$, is the unique graph having no vertices (hence its order is zero). It follows that ${\displaystyle K_{0}}$ also has no edges. Some authors exclude ${\displaystyle K_{0}}$ from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including ${\displaystyle K_{0}}$ as a valid graph is useful depends on context. On the positive side, ${\displaystyle K_{0}}$ 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 ${\displaystyle K_{0}}$ 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 ${\displaystyle K_{0}}$ 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.

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