The command Lovasz(r) returns the Lovasz hypergraph of rank r.

A formal definition  can be found on P. 58 in the English version (available online) of the book of Claude Berge  referenced below.

$\mathrm{with}\left(\mathrm{Hypergraphs}\right)\:$$\mathrm{with}\left(\mathrm{ExampleHypergraphs}\right)\:$

$H≔\mathrm{Lovasz}\left(4\right)$

$H≔\mathrm{<\; a\; hypergraph\; on\; 10\; vertices\; with\; 41\; hyperedges\; >}$

$\mathrm{Vertices}\left(H\right)\&semi;$$\mathrm{Hyperedges}\left(H\right)$

$\left[1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\&comma;9\&comma;10\right]$

$\left[\left\{1\&comma;2\&comma;4\&comma;7\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;7\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;7\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;7\right\}\&comma;\left\{1\&comma;3\&comma;5\&comma;7\right\}\&comma;\left\{2\&comma;3\&comma;5\&comma;7\right\}\&comma;\left\{1\&comma;2\&comma;6\&comma;7\right\}\&comma;\left\{1\&comma;3\&comma;6\&comma;7\right\}\&comma;\left\{2\&comma;3\&comma;6\&comma;7\right\}\&comma;\left\{4\&comma;5\&comma;6\&comma;7\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;8\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;8\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;8\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;8\right\}\&comma;\left\{1\&comma;3\&comma;5\&comma;8\right\}\&comma;\left\{2\&comma;3\&comma;5\&comma;8\right\}\&comma;\left\{1\&comma;2\&comma;6\&comma;8\right\}\&comma;\left\{1\&comma;3\&comma;6\&comma;8\right\}\&comma;\left\{2\&comma;3\&comma;6\&comma;8\right\}\&comma;\left\{4\&comma;5\&comma;6\&comma;8\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;9\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;9\right\}\&comma;\left\{1\&comma;3\&comma;5\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;5\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;6\&comma;9\right\}\&comma;\left\{1\&comma;3\&comma;6\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;6\&comma;9\right\}\&comma;\left\{4\&comma;5\&comma;6\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;10\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;5\&comma;10\right\}\&comma;\left\{2\&comma;3\&comma;5\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;6\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;6\&comma;10\right\}\&comma;\left\{2\&comma;3\&comma;6\&comma;10\right\}\&comma;\left\{4\&comma;5\&comma;6\&comma;10\right\}\&comma;\left\{7\&comma;8\&comma;9\&comma;10\right\}\right]$

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

$G≔\mathrm{LineGraph}\left(H\right)\&semi;$$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(G\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 41\; vertices,\; 820\; edge(s),\; and\; 41\; self-loop(s)}$

$T≔\mathrm{Transversal}\left(H\right)$

$T≔\mathrm{<\; a\; hypergraph\; on\; 10\; vertices\; with\; 41\; hyperedges\; >}$

$\mathrm{AreEqual}\left(H\&comma;T\right)$

$\mathrm{true}$

Claude Berge. Hypergraphes. Combinatoires des ensembles finis. 1987,  Paris, Gauthier-Villars, translated to English.

Claude Berge. Hypergraphs. Combinatorics of Finite Sets.  1989, Amsterdam, North-Holland Mathematical Library, Elsevier, translated from French.

Charles Leiserson, Liyun Li, Marc Moreno Maza and Yuzhen Xie " Parallel computation of the minimal elements of a poset." Proceedings of the 4th International Workshop on Parallel Symbolic Computation (PASCO) 2010: 53-62, ACM.

The Hypergraphs[ExampleHypergraphs][Lovasz] command was introduced in Maple 2024.

For more information on Maple 2024 changes, see Updates in Maple 2024.