Cycle graph

Last updated
Cycle graph
Undirected 6 cycle.svg
A cycle graph of length 6
Vertices n
Edges n
Girth n
Automorphisms 2n (Dn)
Chromatic number 3 if n is odd
2 otherwise
Chromatic index 3 if n is odd
2 otherwise
Spectrum {2 cos(2kπ/n); k = 1, ..., n}  [1]
Properties 2-regular
Vertex-transitive
Edge-transitive
Unit distance
Hamiltonian
Eulerian
Notation
Table of graphs and parameters

In 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 (at least 3, if the graph is simple) 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.

Contents

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. The term n-cycle is sometimes used in other settings. [2]

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:

In addition:

Similarly to the Platonic graphs, the cycle graphs form the skeletons of the dihedra. Their duals are the dipole graphs, which form the skeletons of the hosohedra.

Directed cycle graph

A directed cycle graph of length 8 DC8.png
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

Related Research Articles

Cycle (graph theory) in graph theory, path of edges and vertices wherein a vertex is reachable from itself

In graph theory, a cycle in a graph is a non-empty trail in which the only repeated vertices are the first and last vertices. A directed cycle in a directed graph is a non-empty directed trail in which the only repeated vertices are the first and last vertices.

Bipartite graph graph of two disjoint sets in which every vertex in one set is connected to at least one in the other

In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

Hamiltonian path path in a graph that visits each vertex exactly once

In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete.

This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.

Eulerian path trail in a finite graph which visits every edge exactly once

In graph theory, an Eulerian trail is a trail in a finite graph that visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this:

In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).

Edge coloring an assignment of colors to the edges of a graph so that no two edges that share an endpoint have the same color as each other

In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.

Cross-polytope regular polytope dual to the hypercube in any number of dimensions

In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensions. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

Cubic graph node-link graphs in which every vertex is incident to exactly three edges

In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.

Degree (graph theory) number of edges incident to a given vertex in a node-link graph

In graph theory, the degree of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex is denoted or . The maximum degree of a graph , denoted by , and the minimum degree of a graph, denoted by , are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.

Desargues graph highly symmetric graph with 20 vertices and 30 edges

In the mathematical field of graph theory, the Desargues graph is a distance-transitive cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level of symmetry, is the only known non-planar cubic partial cube, and has been applied in chemical databases.

Graph factorization

In graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is an edge coloring with k colors. A 2-factor is a collection of cycles that spans all vertices of the graph.

Circulant graph Undirected graph acted on by a vertex-transitive cyclic group of symmetries

In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term has other meanings.

Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different contexts.

Rooks graph graph that represents all legal moves of the rook chess piece on a chessboard

In graph theory, a rook's graph is a graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and each edge represents a legal move from one square to another. The same graphs can be defined mathematically as the Cartesian products of two complete graphs or as the line graphs of complete bipartite graphs.

A thrackle is an embedding of a graph in the plane, such that each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. In the latter case, the crossing must be transverse.

In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs.

Odd graph

In the mathematical field of graph theory, the odd graphsOn are a family of symmetric graphs with high odd girth, defined from certain set systems. They include and generalize the Petersen graph.

Pancyclic graph

In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph. Pancyclic graphs are a generalization of Hamiltonian graphs, graphs which have a cycle of the maximum possible length.

In the mathematical field of graph theory, a prism graph is a graph that has one of the prisms as its skeleton.

References

  1. Some simple graph spectra. win.tue.nl
  2. "Problem 11707". Amer. Math. Monthly. 120 (5): 469–476. May 2013. doi:10.4169/amer.math.monthly.120.05.469. JSTOR   10.4169/amer.math.monthly.120.05.469.