Cycle graph

This article is about connected, 2-regular graphs. For other uses, see Cycle graph (disambiguation).

Cycle graph
A cycle graph of length 6
Vertices	n
Edges	n
Girth	n
Automorphisms	2n (Dn)
Chromatic number	3 if n is odd 2 if n is even
Chromatic index	3 if n is odd 2 if n is even
Spectral Gap	$\frac{1-\cos\left(\frac{2 \pi}{n}\right)}{2}$
Properties	2-regular Vertex-transitive Edge-transitive Unit distance Hamiltonian Eulerian
Notation	Cn
v · graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called Cn. The number of vertices in Cn equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it. Terminology There are many synonyms for "cycle graph". These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or n-gon are also often used. A cycle with an even number of vertices is called an even cycle; a cycle with an odd number of vertices is called an odd cycle. Properties A cycle graph is: Connected 2-regular Eulerian Hamiltonian 2-vertex colorable, if and only if it has an even number of vertices. More generally, a graph is bipartite if and only if it has no odd cycles (Kőnig, 1936). 2-edge colorable, if and only if it has an even number of vertices 3-vertex colorable and 3-edge colorable, for any number of vertices A unit distance graph In addition: As cycle graphs can be drawn as regular polygons, the symmetries of an n-cycle are the same as those of a regular polygon with n sides, the dihedral group of order 2n. In particular, there exist symmetries taking any vertex to any other vertex, and any edge to any other edge, so the n-cycle is a symmetric graph. Directed cycle graph A directed cycle graph of length 8 A directed cycle graph is a directed version of a cycle graph, with all the edges being oriented in the same direction. In a directed graph, a set of edges which contains at least one edge (or arc) from each directed cycle is called a feedback arc set. Similarly, a set of vertices containing at least one vertex from each directed cycle is called a feedback vertex set. A directed cycle graph has uniform in-degree 1 and uniform out-degree 1. Directed cycle graphs are Cayley graphs for cyclic groups (see e.g. Trevisan). See also Complete bipartite graph Path graph Complete graph Null graph External links Weisstein, Eric W., "Cycle Graph" from MathWorld. (discussion of both 2-regular cycle graphs and the group-theoretic concept of cycle diagrams) Luca Trevisan, Characters and Expansion. Wikimedia Foundation. 2010. Игры ⚽ Нужно сделать НИР? Leon Wieseltier Sleepover Look at other dictionaries: Cycle graph (algebra) — For other uses, see Cycle graph (disambiguation). In group theory, a sub field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. For… …   Wikipedia Cycle (graph theory) — In graph theory, the term cycle may refer to a closed path. If repeated vertices are allowed, it is more often called a closed walk. If the path is a simple path, with no repeated vertices or edges other than the starting and ending vertices, it… …   Wikipedia Cycle graph (disambiguation) — A cycle graph or cyclic graph is a connected, 2 regular graph.Cycle graph or cyclic graph may also refer to: * Cycle graph (algebra), a diagram representing the cycles determined by taking powers of group elements * Circulant graph * A graph… …   Wikipedia Graph (mathematics) — This article is about sets of vertices connected by edges. For graphs of mathematical functions, see Graph of a function. For statistical graphs, see Chart. Further information: Graph theory A drawing of a labeled graph on 6 vertices and 7 edges …   Wikipedia Graph homomorphism — Not to be confused with graph homeomorphism. In the mathematical field of graph theory a graph homomorphism is a mapping between two graphs that respects their structure. More concretely it maps adjacent vertices to adjacent vertices. Contents 1… …   Wikipedia Cycle — Contents 1 Chemistry 2 Economics 3 Mathematics 4 Music …   Wikipedia Cycle (mathematics) — This article is about group theory. For cycles in homological algebra, see Chain complex#Fundamental terminology. For cycles in graph theory, see Cycle (graph theory). In mathematics, and in particular in group theory, a cycle is a permutation of …   Wikipedia Cycle double cover — Unsolved problems in mathematics Does every bridgeless graph have a multiset of cycles covering every edge exactly twice? …   Wikipedia Cycle space — This article is about a concept in graph theory. For space allocated to bicycles, see segregated cycle facilities. In graph theory, an area of mathematics, a cycle space is a vector space defined from an undirected graph; elements of the cycle… …   Wikipedia Graph coloring — A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible. In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called colors to elements of a graph… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”