StarGraph

The star graph on $n$ vertices — one centre joined to $n - 1$ leaves.

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StarGraph(n)vertex 1 is the centre, joined to vertices

2, \dots, n

Typed signatures

Type	Package	Applies
`(integer) -> value`	`combinatorics`	always

Domain: Collections

Examples

Basic #

The centre of a 6-vertex star has degree 5

\mathrm{VertexDegree}(\mathrm{StarGraph}(6), 1)

Out

5

Scope #

One hub joined to seven leaves

\mathrm{StarGraph}(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)\bigr\rbrack)

A hub with five spokes

\mathrm{StarGraph}(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)\bigr\rbrack)

Applications #

The hub reaches everything in one step; each leaf needs two

\mathrm{Map}(()↦\mathrm{VertexEccentricity}(\mathrm{StarGraph}(6), \operatorname{\_1}), \mathrm{VertexList}(\mathrm{StarGraph}(6)))

Out

\bigl\lbrack1, 2, 2, 2, 2, 2\bigr\rbrack

Properties #

A star is a tree

\mathrm{IsTreeGraph}(\mathrm{StarGraph}(5))

Out

\mathrm{True}

Hub on one side, leaves on the other

\mathrm{IsBipartiteGraph}(\mathrm{StarGraph}(5))

Out

\mathrm{True}

A hub joined to six leaves

\mathrm{StarGraph}(7)

Out

\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5, 6, 7\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)\bigr\rbrack)

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