IsStronglyRegular - Maple Help

GraphTheory

 IsStronglyRegular
 test if graph is strongly regular

 Calling Sequence IsStronglyRegular(G,opts)

Parameters

 G - graph opts - (optional) equation of the form parameters=true or parameters=false

Options

 • parameters : keyword option of the form parameters=true or parameters=false. This specifies whether the parameters [k, lambda, mu] should be returned when the graph is strongly regular. The default is false.

Description

 • The IsStronglyRegular(G) command returns true if G is a strongly regular graph and false otherwise.

Definition

 • An undirected graph G is strongly regular if there exist integers k, lambda, and mu such that every vertex has k neighbors and for every pair of vertices (u,v), u and v have exactly lambda neighbors in common if they are themselves adjacent, and exactly mu neighbors in common if they are not.
 • Note that some parts of this definition may be satisfied trivially, in which a complete graph every pair of vertices is adjacent, so the choice of mu could be arbitrary and therefore mu is undefined.
 • Any strongly regular graph is regular, but the converse is not true.

Strongly regular graphs in SpecialGraphs

 • The following are graphs in the SpecialGraphs subpackage which are strongly regular.

 Graph Number of Vertices k lambda mu 6 4 2 4 10 3 0 1 16 5 0 2 16 6 2 2 27 16 10 8 50 7 0 1 56 10 0 2 77 16 0 4 81 20 1 6 100 22 0 6 231 30 9 3 243 22 1 2 275 112 30 56 1782 416 100 96

Examples

 > $\mathrm{with}\left(\mathrm{GraphTheory}\right):$
 > $\mathrm{with}\left(\mathrm{SpecialGraphs}\right):$
 > $G≔\mathrm{Graph}\left(\left\{\left\{1,2\right\},\left\{1,3\right\},\left\{2,3\right\},\left\{3,4\right\}\right\}\right)$
 ${G}{≔}{\mathrm{Graph 1: an undirected unweighted graph with 4 vertices and 4 edge\left(s\right)}}$ (1)
 > $\mathrm{DegreeSequence}\left(G\right)$
 $\left[{2}{,}{2}{,}{3}{,}{1}\right]$ (2)
 > $\mathrm{IsStronglyRegular}\left(G\right)$
 ${\mathrm{false}}$ (3)
 > $P≔\mathrm{PetersenGraph}\left(\right)$
 ${P}{≔}{\mathrm{Graph 2: an undirected unweighted graph with 10 vertices and 15 edge\left(s\right)}}$ (4)
 > $\mathrm{DegreeSequence}\left(P\right)$
 $\left[{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}\right]$ (5)
 > $\mathrm{IsStronglyRegular}\left(P,'\mathrm{parameters}'\right)$
 ${\mathrm{true}}{,}\left[{3}{,}{0}{,}{1}\right]$ (6)
 > $\mathrm{DrawGraph}\left(P\right)$
 > $C≔\mathrm{ClebschGraph}\left(\right)$
 ${C}{≔}{\mathrm{Graph 3: an undirected unweighted graph with 16 vertices and 40 edge\left(s\right)}}$ (7)
 > $\mathrm{IsStronglyRegular}\left(C,'\mathrm{parameters}'\right)$
 ${\mathrm{true}}{,}\left[{5}{,}{0}{,}{2}\right]$ (8)
 > $\mathrm{DrawGraph}\left(C\right)$

Compatibility

 • The GraphTheory[IsStronglyRegular] command was introduced in Maple 2019.