In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, the term "null graph" may refer either to the order-

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

, 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 In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegr ...

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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

definitions of a graph (it is the ordered pair for which the vertex and edge sets, and , are both

empty Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...

), in proofs it serves as a natural base case for

mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...

, and similarly, in recursively defined data structures is useful for defining the base case for recursion (by treating the null

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

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 In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph.. Definitions While graph drawing an ...

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 category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

in the category. does fulfill ( vacuously) most of the same basic graph properties as does (the graph with one vertex and no edges). As some examples, is of

size Size in general is the magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions ( length, width, height, diameter, perimeter), area, or volume. Size can also be m ...

zero, it is equal to its

complement graph In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of ...

, a

forest A forest is an area of land dominated by trees. Hundreds of definitions of forest are used throughout the world, incorporating factors such as tree density, tree height, land use, legal standing, and ecological function. The United Nations' ...

, and a planar graph. It may be considered

undirected In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...

directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...

, or even both; when considered as directed, it is a

directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one v ...

. And it is both a

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...

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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...

, 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. The notation arises from the fact that the -vertex edgeless graph is the

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...

of the

Notes

References

{{commons category, Null graphs * 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. Graph families Regular graphs

Order-zero graph

Edgeless graph

See also

Notes

References