parameters : keyword option of the form parameters=true or parameters=false. This specifies whether the parameters [d, lambda] should be returned when the graph is Ramanujan. The default is false.

The IsRamanujanGraph(G) command returns true if G is a Ramanujan graph and false otherwise.

An undirected graph G is Ramanujan if it is a regular graph with degree d and $\mathrm{lambda}\le 2\sqrt{d-1}$, where $\mathrm{lambda}$ is the maximum of the absolute values of all the eigenvalues of the adjacency matrix of G excluding the greatest eigenvalue d.

These graphs are named after Srinivasa Ramanujan.

The following are graphs in the SpecialGraphs subpackage which are Ramanujan.

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

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

$G≔\mathrm{Graph}\left(\left\{\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{2\,3\right\}\,\left\{3\,4\right\}\right\}\right)$

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

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

$\mathrm{false}$

$P≔\mathrm{PetersenGraph}\left(\right)$

$P≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 10\; vertices\; and\; 15\; edge(s)}$

$\mathrm{IsRamanujanGraph}\left(P\,'\mathrm{parameters}'\right)$

$\mathrm{true},\left[3\,2.000000000\right]$

$P≔\mathrm{PaleyGraph}\left(29\right)$

$P≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 29\; vertices\; and\; 203\; edge(s)}$

$\mathrm{IsRamanujanGraph}\left(P\,'\mathrm{parameters}'\right)$

$\mathrm{true},\left[14\,3.192582400\right]$

$C≔\mathrm{ClebschGraph}\left(\right)$

$C≔\mathrm{Graph\; 4:\; an\; undirected\; graph\; with\; 16\; vertices\; and\; 40\; edge(s)}$

$\mathrm{IsRamanujanGraph}\left(C\,'\mathrm{parameters}'\right)$

$\mathrm{true},\left[5\,3.000000000\right]$

The GraphTheory[IsRamanujanGraph] command was introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.