|Order-zero graph (null graph)|
The order-zero graph, , is the unique graph having no vertices (hence its order is zero). It follows that also has no edges. 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 (V, E) for which the vertex and edge sets, V and E, are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures 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, "counting all strongly connected components of a graph" becomes "counting all non-null strongly connected components of a graph"). 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.
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 zero, it is equal to its complement graph , it is a connected component, 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 .
|Edgeless graph (empty graph, null graph)|
For each natural number n, the edgeless graph (or empty graph) of order n is the graph with 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.
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- 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.