The GraphJoin(G,H) function returns the graph which is obtained by connecting all the vertices of G to all vertices of H. The vertex labels in the resulting graph are strings of the form "1:u" and "2:v" where u is a vertex in G and v is a vertex in H.

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

$G≔\mathrm{GraphJoin}\left(\mathrm{PathGraph}\left(2\right)\,\mathrm{Graph}\left(3\right)\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 7\; edge(s)}$

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

$\left[''1:1''\,''1:2''\,''2:1''\,''2:2''\,''2:3''\right]$

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

$\left\{\left\{''1:1''\,''1:2''\right\}\,\left\{''1:1''\,''2:1''\right\}\,\left\{''1:1''\,''2:2''\right\}\,\left\{''1:1''\,''2:3''\right\}\,\left\{''1:2''\,''2:1''\right\}\,\left\{''1:2''\,''2:2''\right\}\,\left\{''1:2''\,''2:3''\right\}\right\}$