GraphTheory/IsBiregular - Maple Help

GraphTheory

 IsBiregular
 test if graph is biregular

 Calling Sequence IsBiregular(G) IsBiregular(G, P)

Parameters

 G - graph

Options

 The options argument can contain one or more of the options shown below.
 • partition=truefalse
 If partition=true and G is biregular, two lists of vertices comprising a biregular partition of G are returned. Otherwise a simple Boolean value is returned indicating whether the graph is biregular.

Description

 • IsBiregular returns true if the graph G is biregular and false otherwise. If a variable name P is specified, then this name is assigned a bipartition of the vertices as a list of lists.
 • A graph G is biregular if its set of vertices can be partitioned into two sets, ${V}_{1}$ and ${V}_{2}$, such that every edge in G connects a vertex in ${V}_{1}$ to a vertex in ${V}_{2}$ and if there exist nonnegative integers ${\mathrm{D}}_{1}$ and ${\mathrm{D}}_{2}$ such that every vertex in ${V}_{1}$ has degree ${\mathrm{D}}_{1}$ and every vertex in ${V}_{2}$ has degree ${\mathrm{D}}_{2}$.

Examples

 > $\mathrm{with}\left(\mathrm{GraphTheory}\right):$
 > $\mathrm{K32}≔\mathrm{CompleteGraph}\left(3,2\right)$
 ${\mathrm{K32}}{≔}{\mathrm{Graph 1: an undirected unweighted graph with 5 vertices and 6 edge\left(s\right)}}$ (1)
 > $\mathrm{IsBiregular}\left(\mathrm{K32},'\mathrm{partition}'\right)$
 ${\mathrm{true}}{,}\left[{1}{,}{2}{,}{3}\right]{,}\left[{4}{,}{5}\right]$ (2)
 > $\mathrm{DrawGraph}\left(\mathrm{K32},\mathrm{style}=\mathrm{bipartite}\right)$
 > $\mathrm{AdjacencyMatrix}\left(\mathrm{K32}\right)$
 $\left[\begin{array}{ccccc}{0}& {0}& {0}& {1}& {1}\\ {0}& {0}& {0}& {1}& {1}\\ {0}& {0}& {0}& {1}& {1}\\ {1}& {1}& {1}& {0}& {0}\\ {1}& {1}& {1}& {0}& {0}\end{array}\right]$ (3)
 > $G≔\mathrm{CycleGraph}\left(5\right)$
 ${G}{≔}{\mathrm{Graph 2: an undirected unweighted graph with 5 vertices and 5 edge\left(s\right)}}$ (4)
 > $\mathrm{IsBiregular}\left(G\right)$
 ${\mathrm{false}}$ (5)

Compatibility

 • The GraphTheory[IsBiregular] command was introduced in Maple 2019.