In the case of undirected graphs, only O ( n ) time is required, since at most n − 1 edges can be tree edges ( where n is the number of vertices ).

The number of different Hamiltonian cycles in a complete undirected graph on n vertices is and in a complete directed graph on n vertices is.

The number of vertices must be doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph.

Another theorem that is commonly proved with a double counting argument states that every undirected graph contains an even number of vertices of odd degree.

Counting the number of Eulerian circuits on undirected graphs is much more difficult.

* The handshaking lemma, proven by Euler in his original paper, showing that any undirected connected graph has an even number of odd-degree vertices

2-satisfiability has also been applied to problems of recognizing undirected graphs that can be partitioned into an independent set and a small number of complete bipartite subgraphs, inferring business relationships among autonomous subsystems of the internet, and reconstruction of evolutionary trees.

* In the independent set decision problem, the input is an undirected graph and a number k, and the output is a Boolean value: true if the graph contains an independent set of size k, and false otherwise.

What is the maximum possible number of edges in an undirected graph G with n vertices which does not contain K < sub > 3 </ sub > ( three vertices A, B, C with edges AB, AC, BC ; i. e. a triangle ) as a subgraph?

A perfect graph is an undirected graph with the property that, in every one of its induced subgraphs, the size of the largest clique equals the minimum number of colors in a coloring of the subgraph.

An undirected graph is perfect if, in every induced subgraph, the chromatic number equals the size of the largest clique.

In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph have nonzero linking number.

* Do two undirected graphs have the same number of connected components?

For an undirected graph, the degree of a vertex is equal to the number of adjacent vertices.

The BEST theorem shows that the number of Eulerian circuits in directed graphs can be computed in polynomial time, a problem which is # P-complete for undirected graphs.

Let be the number of triangles on for undirected graph.

The betweenness may be normalised by dividing through the number of pairs of vertices not including v, which for directed graphs is and for undirected graphs is.

In graph-theoretic mathematics, the circuit rank or cyclomatic number of an undirected graph is the minimum number of edges to remove from to remove all its cycles, making it into a forest.

* The metric dimension of an undirected graph G is the minimum number of vertices in a subset S of G such that all other vertices are uniquely determined by their distances to the vertices in S.

In an unweighted undirected graph, the size or weight of a cut is the number of edges crossing the cut.

The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned.

However, the rail link between Abkhazia and Georgia proper has been closed for a number of years, forcing Armenia to receive rail cars laden with cargo only through the relatively expensive rail-ferry services operating between Georgian and other Black Sea ports.

Many branch lines ( and a number of main lines ) were closed because they were deemed uneconomic (" the Beeching Axe " of 1963 ), removing much feeder traffic from main line passenger services.

Most motorways are covered by the closed toll collection system, where a driver receives a ticket at the entrance gates and pays at the exit gates according to the number of sections travelled.

Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space.

In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary.

* Consider the set of all functions from the real number line to the closed unit interval, and define a topology on so that a sequence in converges towards if and only if converges towards for all.

On March 16, 1517, the Fifth Council of the Lateran closed its activities with a number of reform proposals ( on the selection of bishops, taxation, censorship and preaching ) but not on the major problems that confronted the Church in Germany and other parts of Europe.

" In Golden Age detective stories, an outsider — sometimes a salaried investigator or a police officer, but often a gifted amateur — investigates a murder committed in a closed environment by one of a limited number of suspects.

By 1968, the number of dance clubs started to decline ; most American clubs either closed or were transformed into clubs featuring live bands.

As a consequence of this, numerous German production and distribution companies went out of business in the 1950s and 1960s and cinemas across the Federal Republic closed their doors ; the number of screens in West Germany almost halved between the beginning and the end of the decade.

Distressed by the increasing number of closed theatres, studios and companies would find new and innovative ways to bring audiences back.

For closed ( orientable or non-orientable ) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

Their number increased significantly during the 18th century, and again when many Spaniards left their jobs in Gibraltar after General Franco closed the border in 1969.

After the end of World War I, the majority of the United Kingdom gunpowder manufacturers merged into a single company, " Explosives Trades limited "; and number of sites were closed down, including those in Ireland.

He appointed a commission that set aside 3, 000, 000 acres ( 12, 000 km² ) of national parks and 2, 300, 000 acres of national forests ; advocated tax reduction for low-income Americans ( not enacted ); closed certain tax loopholes for the wealthy ; doubled the number of veterans ' hospital facilities ; negotiated a treaty on St. Lawrence Seaway ( which failed in the U. S. Senate ); wrote a Children's Charter that advocated protection of every child regardless of race or gender ; created an antitrust division in the Justice Department ; required air mail carriers to adopt stricter safety measures and improve service ; proposed federal loans for urban slum clearances ( not enacted ); organized the Federal Bureau of Prisons ; reorganized the Bureau of Indian Affairs ; instituted prison reform ; proposed a federal Department of Education ( not enacted ); advocated $ 50-per-month pensions for Americans over 65 ( not enacted ); chaired White House conferences on child health, protection, homebuilding and home-ownership ; began construction of the Boulder Dam ( later renamed Hoover Dam ); and signed the Norris – La Guardia Act that limited judicial intervention in labor disputes.

Whether these initiatives will have the desired effect of curbing such activities has yet to be seen ; although a number of hawala networks have been closed down and a number of hawaladars have been successfully prosecuted for money laundering, there is little sign that these actions have brought the authorities any closer to identifying and arresting a significant number of terrorists or drug smugglers.

These procedures consist in assuming that he can isolate from the rest of the world a closed system containing a limited number of variables and a limited range of consequences .”

Communication systems used by other animals such as bees or non-human apes are closed systems that consist of a closed number of possible things that can be expressed.

' Counting ' the number of field lines in a closed surface, therefore, yields the total charge enclosed by that surface.

* Meander ( mathematics ), a self-avoiding closed curve which intersects a line a number of times

The game starts with any number of players and a number of chairs one fewer than the number of players ; the chairs are arranged in a circle ( or other closed figure if space is constrained ; a double line is sometimes used ) facing outward, with the people standing in a circle just outside of that.