The CageGraph(K,G) command creates the (K,G)-cage graph, that is, the smallest K-regular graph(s) with girth G.

If more than one graph meets these criteria, a sequence of graphs is returned.

In the cases where the smallest graph is not known, FAIL is returned.

Several notable cage graphs have been assigned names and can be individually accessed under these names within the SpecialGraphs subpackage. These include:

(3,5)-cage: PetersenGraph

(3,6)-cage: HeawoodGraph

(3,8)-cage: Tutte8CageGraph

(3,10)-cages: Balaban10CageGraph, HarriesGraph, and HarriesWongGraph

(3,12)-cage: GeneralizedHexagonGraph

(4,5)-cage: RobertsonGraph

(5,5)-cages: FosterCageGraph, MeringerGraph, RobertsonWegnerGraph, and WongGraph

(7,5)-cage: HoffmanSingletonGraph

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

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

$C≔\mathrm{CageGraph}\left(3\,5\right)$

$C≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 10\; vertices\; and\; 15\; edge(s)}$

$\mathrm{DrawGraph}\left(C\right)$

$\mathrm{C3}≔\mathrm{CageGraph}\left(3\,10\right)$

$\mathrm{C3}≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 70\; vertices\; and\; 105\; edge(s)},\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 70\; vertices\; and\; 105\; edge(s)},\mathrm{Graph\; 4:\; an\; undirected\; graph\; with\; 70\; vertices\; and\; 105\; edge(s)}$

$\mathrm{CageGraph}\left(8\,11\right)$

$\mathrm{FAIL}$