CycleGraph

The cycle graph $C_n$ — $n$ vertices joined in a ring.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageCycleGraph!↗

CycleGraph(n)$C_n$ ($n \geq 3$), vertices $1, \dots, n$, each joined to its neighbours around the ring.

SageMathenumeratio_graph(graphs.CycleGraph(n))

Typed signatures

Type	Package	Applies
(integer) -> value	combinatorics	always

Domain: Collections

Examples

Basic #

An odd cycle is never bipartite

live ↗

In$\mathrm{IsBipartiteGraph}(\mathrm{CycleGraph}(5))$Out$\mathrm{False}$

An even cycle is bipartite

live ↗

In$\mathrm{IsBipartiteGraph}(\mathrm{CycleGraph}(6))$Out$\mathrm{True}$

Scope #

The 3-cycle is a triangle

live ↗

In$\mathrm{CycleGraph}(3)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(3, 1)\bigr\rbrack)$

Applications #

On a 5-cycle the farthest vertex is 2 steps away

live ↗

In$\mathrm{Map}(\operatorname{\_1}↦\mathrm{VertexEccentricity}(\mathrm{CycleGraph}(5), \operatorname{\_1}), \mathrm{VertexList}(\mathrm{CycleGraph}(5)))$Out$\bigl\lbrack2, 2, 2, 2, 2\bigr\rbrack$

Properties #

A cycle graph contains a cycle, so not acyclic

live ↗

In$\mathrm{IsAcyclicGraph}(\mathrm{CycleGraph}(5))$Out$\mathrm{False}$

Every vertex has degree 2

live ↗

In$\mathrm{VertexDegree}(\mathrm{CycleGraph}(7))$Out$\bigl\lbrack2, 2, 2, 2, 2, 2, 2\bigr\rbrack$

Four vertices joined in a ring

live ↗

In$\mathrm{CycleGraph}(4)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(4, 1)\bigr\rbrack)$

live ↗

In$\mathrm{IsPathGraph}(\mathrm{CycleGraph}(5))$Out$\mathrm{False}$

About

Examples	8
Checked against	Wolfram Language 7 agree1 differSageMath 4 agree2 differ
Crosswalk	Wolfram LanguageSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

See also: PathGraph, IsBipartiteGraph