WheelGraph

The wheel graph on n vertices — a hub joined to every vertex of an (n-1)-cycle rim.

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

WheelGraph(n)vertex 1 is the hub; vertices 2..n form the rim cycle. Needs n >= 4.

SageMathenumeratio_graph(graphs.WheelGraph(n))

Typed signatures

Type	Package	Applies
(integer) -> value	combinatorics	always

Domain: Collections

Details

• EdgeCount = 2(n - 1): n - 1 spokes plus the n - 1 rim edges. Vertex numbering (which vertex is the hub) is kernel-unverified against a live Wolfram session.

Examples

Basic #

live ↗

In$\mathrm{EdgeCount}(\mathrm{WheelGraph}(6))$Out$10$

Scope #

A hub joined to a $7$-cycle

live ↗

In$\mathrm{WheelGraph}(8)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5, 6, 7, 8\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(1, 5), \mathrm{UndirectedEdge}(1, 6), \mathrm{UndirectedEdge}(1, 7), \mathrm{UndirectedEdge}(1, 8), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(4, 5), \mathrm{UndirectedEdge}(5, 6), \mathrm{UndirectedEdge}(6, 7), \mathrm{UndirectedEdge}(7, 8), \mathrm{UndirectedEdge}(8, 2)\bigr\rbrack)$

A hub plus a $4$-cycle rim

live ↗

In$\mathrm{WheelGraph}(5)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(1, 5), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(4, 5), \mathrm{UndirectedEdge}(5, 2)\bigr\rbrack)$

A hub joined to a $5$-cycle

live ↗

In$\mathrm{WheelGraph}(6)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(1, 5), \mathrm{UndirectedEdge}(1, 6), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(4, 5), \mathrm{UndirectedEdge}(5, 6), \mathrm{UndirectedEdge}(6, 2)\bigr\rbrack)$

Applications #

The hub reaches everything in one step; rim vertices need two

live ↗

In$\mathrm{Map}(\operatorname{\_1}↦\mathrm{VertexEccentricity}(\mathrm{WheelGraph}(6), \operatorname{\_1}), \mathrm{VertexList}(\mathrm{WheelGraph}(6)))$Out$\bigl\lbrack1, 2, 2, 2, 2, 2\bigr\rbrack$

Properties #

A hub with a triangle rim makes the complete graph on four vertices

live ↗

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

The wheel on $4$ vertices coincides with $K_4$

live ↗

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

About

Examples	7
Checked against	Wolfram Language 7 agreeSageMath 1 agree5 differ
Crosswalk	Wolfram LanguageSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

See also: CycleGraph, StarGraph