Dense graph

In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem.

The graph density of simple graphs is defined to be the ratio of the number of edges |E| with respect to the maximum possible edges.

For undirected simple graphs, the graph density is:

${\displaystyle D={\frac {|E|}{\binom {|V|}{2}}}={\frac {2|E|}{|V|(|V|-1)}}}$

For directed, simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is:

${\displaystyle D={\frac {|E|}{2{\binom {|V|}{2}}}}={\frac {|E|}{|V|(|V|-1)}}}$

where E is the number of edges and V is the number of vertices in the graph. The maximum number of edges for an undirected graph is ${\displaystyle {\binom {|V|}{2}}={\frac {|V|(|V|-1)}{2}}}$, so the maximal density is 1 (for complete graphs) and the minimal density is 0 (Coleman & Moré 1983).

For families of graphs of increasing size, one often calls them sparse if ${\displaystyle D\rightarrow 0}$ as ${\displaystyle |V|\rightarrow \infty }$. Sometimes, in computer science, a more restrictive definition of sparse is used like ${\displaystyle |E|=O(|V|\log |V|)}$ or even ${\displaystyle |E|=O(|V|)}$.