Null graph
{{Short description|Order-zero graph or any edgeless 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
{{infobox graph
| name = Order-zero graph (null graph)
| vertices = 0
| edges = 0
| girth = ∞
| automorphisms = 1
| chromatic_number = 0
| chromatic_index = 0
| genus = 0
| spectral_gap = undefined
| notation = {{math|K{{sub|0}}}}
| properties = Integral
Symmetric
Treewidth -1
}}
The order-zero graph, {{math|K{{sub|0}}}}, is the unique graph having no vertices (hence its order is zero). It follows that {{math|K{{sub|0}}}} also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude {{math|K{{sub|0}}}} from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including {{math|K{{sub|0}}}} as a valid graph is useful depends on context. On the positive side, {{math|K{{sub|0}}}} follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair {{math|(V, E)}} for which the vertex and edge sets, {{mvar|V}} and {{mvar|E}}, are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures {{math|K{{sub|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 {{math|K{{sub|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 {{math|K{{sub|0}}}}). 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.{{MathWorld |urlname=EmptyGraph |title=Empty Graph}}{{MathWorld |urlname=NullGraph |title=Null Graph}}
In category theory, the order-zero graph is, according to some definitions of "category of graphs," the initial object in the category.
{{math|K{{sub|0}}}} does fulfill (vacuously) most of the same basic graph properties as does {{math|K{{sub|1}}}} (the graph with one vertex and no edges). As some examples, {{math|K{{sub|0}}}} is of size zero, it is equal to its complement graph {{math|{{overline|K}}{{sub|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 {{math|K{{sub|0}}}}.
Edgeless graph
{{infobox graph
| name = Edgeless graph (empty graph, null graph)
| vertices = {{mvar|n}}
| edges = 0
| radius = 0
| diameter = 0
| girth = ∞
| automorphisms = {{math|n!}}
| chromatic_number = 1
| chromatic_index = 0
| genus = 0
| spectral_gap = undefined
| notation = {{mvar|{{overline|K}}{{sub|n}}}}
| properties = Integral
Symmetric
|Degree=0}}
For each natural number {{mvar|n}}, the edgeless graph (or empty graph) {{mvar|{{overline|K}}{{sub|n}}}} of order {{mvar|n}} is the graph with {{mvar|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.
It is a 0-regular graph. The notation {{mvar|{{overline|K}}{{sub|n}}}} arises from the fact that the {{mvar|n}}-vertex edgeless graph is the complement of the complete graph {{mvar|K{{sub|n}}}}.
See also
Notes
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References
{{refbegin}}
- 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.
{{refend}}