The TetrahedronGraph command creates the tetrahedron graph (the complete graph) on 4 vertices. As an option, you may input the labels of the vertices as a set or list of size 4.

The OctahedronGraph command creates the octahedron graph on 6 vertices. As an option, you may input the labels of the vertices as a set or list of size 6.

The DodecahedronGraph command creates the dodecahedron graph on 20 vertices. A dodecahedron is a 3-regular and 12-faced planar graph. As an option, you may input the labels of the vertices as a set or list of size 20.

The IcosahedronGraph command creates the icosahedron graph on 12 vertices. An icosahedron is a 5-regular and 20-faced planar graph. As an option, you may input the labels of the vertices as a set or list of size 12.

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

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

$T≔\mathrm{TetrahedronGraph}\left(\right)$

$T≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 6\; edge(s)}$

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

$G≔\mathrm{OctahedronGraph}\left(\right)$

$G≔\mathrm{Graph\; 2:\; an\; undirected\; unweighted\; graph\; with\; 6\; vertices\; and\; 12\; edge(s)}$

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

$\mathrm{true}$

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

$H≔\mathrm{DodecahedronGraph}\left(\right)$

$H≔\mathrm{Graph\; 3:\; an\; undirected\; unweighted\; graph\; with\; 20\; vertices\; and\; 30\; edge(s)}$

$\mathrm{Neighborhood}\left(H\,19\right)$

$\left[14\,18\,20\right]$

$\mathrm{IsPlanar}\left(H\,'F'\right)$

$\mathrm{true}$

$\mathrm{nops}\left(F\right)$

$12$

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

$K≔\mathrm{IcosahedronGraph}\left(\right)$

$K≔\mathrm{Graph\; 4:\; an\; undirected\; unweighted\; graph\; with\; 12\; vertices\; and\; 30\; edge(s)}$

$\mathrm{IsPlanar}\left(K\,'F'\right)$

$\mathrm{true}$

$\mathrm{map}\left(\mathrm{nops}\,F\right)$

$\left[3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\,3\right]$

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