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.

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 non-negative 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 ${\mathrm{D}}_2$.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{K32}≔\mathrm{CompleteGraph}\left(3\,2\right)$

$\mathrm{K32}≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 6\; edge(s)}$

$\mathrm{IsBiregular}\left(\mathrm{K32}\,'\mathrm{partition}'\right)$

$\mathrm{true},\left[1\,2\,3\right],\left[4\,5\right]$

$\mathrm{DrawGraph}\left(\mathrm{K32}\,\mathrm{style}=\mathrm{bipartite}\right)$

$\mathrm{AdjacencyMatrix}\left(\mathrm{K32}\right)$

$\left[\begin{array}{ccccc}0& 0& 0& 1& 1\\ 0& 0& 0& 1& 1\\ 0& 0& 0& 1& 1\\ 1& 1& 1& 0& 0\\ 1& 1& 1& 0& 0\end{array}\right]$

$G≔\mathrm{CycleGraph}\left(5\right)$

$G≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 5\; edge(s)}$

$\mathrm{IsBiregular}\left(G\right)$

$\mathrm{false}$

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

For more information on Maple 2019 changes, see Updates in Maple 2019.