How many directed graphs

A simple directed graph on nodes may have between 0 and edges. A complete graph in which each edge is bidirected is called a complete directed graph. A directed graph having no symmetric pair of directed edges i. A complete oriented graph i. This gives the counting polynomial for the number of directed graphs with points as. Harary , p. Here, is the floor function , is a binomial coefficient , LCM is the least common multiple , GCD is the greatest common divisor , the sum is over all exponent vectors of the cycle index , and is the coefficient of the term with exponent vector in.

The first few cycle indices are. We wish to assign a value to a flow, equal to the net flow out of the source. Before we prove this, we introduce some new notation. Theorem 5. We next seek to formalize the notion of a "bottleneck'', with the goal of showing that the maximum flow is equal to the amount that can pass through the smallest bottleneck.

Note that a minimum cut is a minimal cut. Lemma 5. Then the value of a maximum flow is equal to the capacity of a minimum cut.

Using the proof of theorem 5. The max-flow, min-cut theorem is true when the capacities are any positive real numbers, though of course the maximum value of a flow will not necessarily be an integer in this case. It is somewhat more difficult to prove; a proof involves limits. We have already proved that in a bipartite graph, the size of a maximum matching is equal to the size of a minimum vertex cover, theorem 4. This turns out to be essentially a special case of the max-flow, min-cut theorem.

Corollary 5. Now the value of a maximum flow is equal to the capacity of a minimum cut. A digraph is connected if the underlying graph is connected.

Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. In the following graph, it is possible to travel from one vertex to any other vertex. In the graph, a vertex should have edges with all other vertices, then it called a complete graph.

In other words, if a vertex is connected to all other vertices in a graph, then it is called a complete graph. A vertex or node of a graph is one of the objects that are connected together. The connections between the vertices are called edges or links.

A graph with 10 vertices or nodes and 11 edges links. For more information about graph vertices, see the network introduction. An acyclic graph is a graph having no graph cycles.