In the mathematical field of graph theory, the term "null graph" may refer either to the

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

- zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph").

Order-zero graph

The order-zero graph, , is the unique graph having no vertices (hence its order is zero). It follows that also has no

edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

. Thus the null graph is a regular graph of degree zero. Some authors exclude from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including as a valid graph is useful depends on context. On the positive side, follows naturally from the usual set-theoretic definitions of a graph (it is the

ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...

for which the vertex and edge sets, and , are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined

data structure In computer science, a data structure is a data organization, management, and storage format that is usually chosen for efficient access to data. More precisely, a data structure is a collection of data values, the relationships among them, a ...

s is useful for defining the base case for recursion (by treating the null tree as the