an empty graph on nodes consists of isolated nodes with no edges. such graphs are sometimes also called edgeless graphs or null graphs (though the term “null graph” is also used to refer in particular to the empty graph on 0 nodes). the empty graph on 0 nodes is (sometimes) called the null graph and the empty graph on 1 node is called the singleton graph. the empty graph on vertices is the graph complement of the complete graph , and is commonly denoted . the notation is apparently also used by some authors (e.g., tyshkevich 2000, fact 2), but not recommended as it conflicts with use of this notation for an odd graph among others. the empty graph on nodes can be generated in the wolfram language as graph[range[n], ] or fromentity[entity[“graph”, “empty”, n]], and precomputed properties of empty graphs are available in the wolfram language using graphdata[“empty”, n].
empty graph format
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empty graph guide
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”). thus the null graph is a regular graph of degree zero. whether including k0 as a valid graph is useful depends on context. on the negative side, including k0 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 k0).
 in category theory, the order-zero graph is, according to some definitions of “category of graphs,” the initial object in the category. as some examples, k0 is of size zero, it is equal to its complement graph k0, a forest, and a planar graph. and it is both a complete graph and an edgeless graph. for each natural number n, the edgeless graph (or empty graph) kn 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.
this is the base of graph theory. a graph (g) is denoted mathematically as g(v, e), where v represents the collection of vertices and e represents the set of edges. the number of elements in a graph can also be used to characterise it. the number of elements in set x is denoted by |x|. as a result, n0 represents an empty graph. yet, the terms ’empty’ and ‘null’ are the most common ones used to describe a graph with no vertices or edges. a complete graph is the opposite of an empty graph. a complete graph is fully connected, whereas an empty one is completely disconnected.
it provides valuable insights on graph behaviour, guiding decisions in a variety of applications and research. in essence, an empty graph is a simple and unstructured graph that lacks both vertices and edges, making it an elementary concept in graph theory that often serves as a starting point when tackling more complex graph structures and properties. the degree of a vertex in an empty one is commonly referred to as 0, even though it is technically undefined because there are no vertices or edges in this graph. a trivial graph is the most basic form of graph, with a single isolated vertex (v) and no edges—a fundamental non-empty graph. an empty one, on the other hand, is a subset of a trivial graph that has the simplest structure conceivable, with no vertices or edges. they have the benefit of providing a simple foundation for understanding more sophisticated graph structures. this is due to the fact that empty graphs represent a complete lack of structure and connections. while empty graphs may not have direct practical uses in many real-world situations, they may be useful in a few specialised cases or applications: empty graphs may not be useful in everyday situations, but they serve as building blocks for more complex graphs. they are essential to graph-related learning and theory.