In the mathematics 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, , is the unique graph having no vertices (hence its order is zero). It follows that also has no edges. 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 theory definitions of a graph (it is the
ordered pair for which the vertex and edge sets, and , are both
empty set), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined
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 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 ). 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.
does fulfill ([[vacuously|vacuous truth]]) most of the same basic graph properties as does (the graph with one vertex and no edges). As some examples, is of size zero, it is equal to its [[complement graph]] , a forest, and a [[planar graph]]. It may be considered [[undirected|undirected graph]], [[directed|directed graph]], 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 .
Edgeless graph
For each
natural number , the
edgeless graph (or empty graph) of order is the graph with 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 graph. The notation arises from the fact that the -vertex edgeless graph is the complement graph of the complete graph .
See also
Notes
-
Frank Harary and Read, R. (1973), "Is the null graph a pointless concept?", Graphs and Combinatorics (Conference, George Washington University), Springer-Verlag, New York, NY.
External links
