We conclude this chapter with a simple example to illustrate further the use of GRAPE.
In this example we construct the Petersen graph P, and its edge graph
(often called line graph) EP. We compute the (global) parameters of
EP, and so verify that EP is distance-regular (see BCN89). We
also show that there is just one orbit of 1-factors of P under the
automorphism group of P (but you should read the documentation of the
function CompleteSubgraphsOfGivenSize
to see exactly what that function
does).
gap> P := Graph( SymmetricGroup(5), [[1,2]], OnSets, > function(x,y) return Intersection(x,y)=[]; end ); rec( isGraph := true, order := 10, group := Group( ( 1, 2)( 6, 8)( 7, 9), ( 1, 3)( 4, 8)( 5, 9), ( 2, 4)( 3, 6)( 9,10), ( 2, 5)( 3, 7)( 8,10) ), schreierVector := [ -1, 1, 2, 3, 4, 3, 4, 2, 2, 4 ], adjacencies := [ [ 8, 9, 10 ] ], representatives := [ 1 ], names := [ [ 1, 2 ], [ 2, 5 ], [ 1, 5 ], [ 2, 3 ], [ 2, 4 ], [ 1, 3 ], [ 1, 4 ], [ 3, 5 ], [ 4, 5 ], [ 3, 4 ] ] ) gap> Diameter(P); 2 gap> Girth(P); 5 gap> EP := EdgeGraph(P); rec( isGraph := true, order := 15, group := Group( ( 1, 4)( 2, 5)( 3, 6)(10,11)(12,13)(14,15), ( 1, 7) ( 2, 8)( 3, 9)(10,15)(11,13)(12,14), ( 2, 3)( 4, 7)( 5,10)( 6,11) ( 8,12)( 9,14), ( 1, 3)( 4,12)( 5, 8)( 6,13)( 7,10)( 9,15) ), schreierVector := [ -1, 3, 4, 1, 3, 1, 2, 3, 2, 4, 1, 4, 1, 2, 2 ], adjacencies := [ [ 2, 3, 13, 15 ] ], representatives := [ 1 ], isSimple := true, names := [ [ [ 1, 2 ], [ 3, 5 ] ], [ [ 1, 2 ], [ 4, 5 ] ], [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 3 ], [ 2, 5 ] ], [ [ 1, 4 ], [ 2, 5 ] ], [ [ 2, 5 ], [ 3, 4 ] ], [ [ 1, 5 ], [ 2, 3 ] ], [ [ 1, 5 ], [ 2, 4 ] ], [ [ 1, 5 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [ [ 2, 3 ], [ 4, 5 ] ], [ [ 1, 3 ], [ 2, 4 ] ], [ [ 2, 4 ], [ 3, 5 ] ], [ [ 1, 3 ], [ 4, 5 ] ], [ [ 1, 4 ], [ 3, 5 ] ] ] ) gap> GlobalParameters(EP); [ [ 0, 0, 4 ], [ 1, 1, 2 ], [ 1, 2, 1 ], [ 4, 0, 0 ] ] gap> CompleteSubgraphsOfGivenSize(ComplementGraph(EP),5); [ [ 1, 5, 9, 11, 12 ] ]
GAP 3.4.4