Null graph

In the mathematical field of graph theory, the null graph or the empty graph is either the graph with no vertices and (hence) no edges, or any graph with no edges.

The null graph (in the former sense) is the initial object in the category of graphs, according to some definitions of a category of graphs. Having no vertices, the null graph therefore also has no connected components. Thus, although the null graph is a forest (a graph with no cycles), it is not a tree, as trees have one connected component.

Edgeless graph

Edgeless graph
Vertices	n
Edges	0
Automorphisms	n!
Chromatic number	1
Properties	Integral Symmetric
Notation
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Some authors feel that a better term for the latter sense—(V, { }) for any set V—is the more explicit edgeless graph. This reserves the term null graph for the former sense: a graph without even any vertices. Still others make this distinction by applying the label empty to these graphs with no edges.^[1][2]

The n-vertex edgeless graph is the complement graph for the complete graph K_n, and therefore it is commonly denoted as .

Even though this definition provides a solid basis for defining certain operations on graphs (eg: decomposition) considering graphs as sets of vertices and edges (V, E) this definition raises a problem in uniqueness of the null element of graphs.

See also

• Glossary of graph theory
• Cycle graph
• Path graph

Notes

References

• 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.

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