|Order-zero graph (null graph)|
The order-zero graph is the unique graph of order zero (having zero vertices). As a consequence, it also has zero edges. In some contexts, is excluded from being considered a graph (either by definition, or more simply as a matter of convenience).
The order-zero graph is the initial object in the category of graphs, according to some definitions of a category of graphs. Its inclusion within the definition of graph theory is more useful in some contexts than others. On the positive side, follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair of empty sets), and 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, most well-defined formulas for graph properties must include exceptions for if it is included as a graph ("counting all strongly connected components of a graph" would become "counting all non-null strongly connected components of a graph"). Due to the undesirable aspects, it is usually assumed in literature that the term "graph" implies "graph with at least one vertex" unless context suggests otherwise.
When acknowledged, fulfills (vacuously) most of the same basic graph properties as (the graph with one vertex and no edges): it has a size of zero, it is equal to its complement graph (), it is a connected component (namely, ), a forest, and a planar graph. It may be an undirected graph or a directed graph (or even both); when considered as directed, it is a directed acyclic graph, and it is both a complete graph and an empty graph (just to name a few). 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) 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.