Quelle grape.xml Sprache: XML

#############################################################################
##
#W grape.xml
#Y Copyright (C) 2014-21 James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##

<#GAPDoc Label="Graph">
<ManSection>
 <Oper Name="Graph" Arg="digraph"/>
 <Returns>A &GRAPE; package graph.</Returns>
 <Description>
 If <A>digraph</A> is a mutable or immutable digraph without multiple edges, then this operation
 returns a &GRAPE; package graph that is isomorphic to <A>digraph</A>. 

 If <A>digraph</A> is a multidigraph, then since &GRAPE; does not support
 multiple edges, the multiple edges will be reduced to a single edge in the
 result. In order words, for a multidigraph this operation will return the
 same as <C>Graph(DigraphRemoveAllMultipleEdges(</C><A>digraph</A><C>))</C>.

 <Example><![CDATA[
gap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,
> function(x, y) return Intersection(x, y) = []; end);;
gap> Display(Petersen);
rec(
 adjacencies := [ [ 3, 5, 8 ] ],
 group :=
 Group( [ ( 1, 2, 3, 5, 7)( 4, 6, 8, 9,10), ( 2, 4)( 6, 9)( 7,10)
 ] ),
 isGraph := true,
 names := [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ],
 [ 2, 4 ], [ 1, 5 ], [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ],
 order := 10,
 representatives := [ 1 ],
 schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )
gap> Digraph(Petersen);
<immutable digraph with 10 vertices, 30 edges>
gap> Graph(last) = Petersen;
true]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="CayleyDigraph">
<ManSection>
 <Oper Name="CayleyDigraph" Arg="[filt,] G[, gens]"/>
 <Returns>A digraph.</Returns>
 <Description>
 Let <A>G</A> be any group and let <A>gens</A> be a list of elements of
 <A>G</A>. This operation returns a digraph that corresponds to the Cayley
 graph of <A>G</A> with respect to <A>gens</A>. 

 The vertices of the digraph correspond to the elements of <A>G</A>, in the
 order given by <C>Set(<A>G</A>)</C>. There exists an edge from vertex
 <C>u</C> to vertex <C>v</C> if and only if there exists a generator
 <C>g</C> in <A>gens</A> such that <C>Set(<A>G</A>)[u] * g =
 Set(<A>G</A>)[v]</C>. 

 The labels of the vertices <C>u</C>, <C>v</C>, and the edge <C>[u, v]</C>
 are the corresponding elements <C>AsList(<A>G</A>)[u]</C>,
 <C>AsList(<A>G</A>)[v]</C>, and generator <C>g</C>, respectively;
 see <Ref Oper="DigraphVertexLabel"/> and <Ref Oper="DigraphEdgeLabel"/>.
 

 &STANDARD_FILT_TEXT;

 If the optional third argument <A>gens</A> is not present, then the
 generators of <A>G</A> are used by default.

 The digraph created by this operation belongs to the category <Ref
 Filt="IsCayleyDigraph"/>, the group <A>G</A> can be recovered from the
 digraph using <Ref Attr="GroupOfCayleyDigraph"/>, and the generators
 <A>gens</A> can be obtained using <Ref Attr="GeneratorsOfCayleyDigraph"/>.

 <Example><![CDATA[
gap> G := DihedralGroup(8);
<pc group of size 8 with 3 generators>
gap> CayleyDigraph(G);
<immutable digraph with 8 vertices, 24 edges>
gap> G := DihedralGroup(IsPermGroup, 8);
Group([ (1,2,3,4), (2,4) ])
gap> CayleyDigraph(G);
<immutable digraph with 8 vertices, 16 edges>
gap> digraph := CayleyDigraph(G, [()]);
<immutable digraph with 8 vertices, 8 edges>
gap> GroupOfCayleyDigraph(digraph) = G;
true
gap> GeneratorsOfCayleyDigraph(digraph);
[ () ]
gap> digraph := CayleyDigraph(IsMutable, G, [()]);
<mutable digraph with 8 vertices, 8 edges>
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="GroupOfCayleyDigraph">
<ManSection>
 <Attr Name="GroupOfCayleyDigraph" Arg="digraph"/>
 <Attr Name="SemigroupOfCayleyDigraph" Arg="digraph"/>
 <Returns>A group or semigroup.</Returns>
 <Description>
 If <A>digraph</A> is an immutable Cayley graph of a group <C>G</C> and
 <A>digraph</A> belongs to the category <Ref Filt="IsCayleyDigraph"/>, then
 <C>GroupOfCayleyDigraph</C> returns <C>G</C>.
 

 If <A>digraph</A> is a Cayley graph of a semigroup <C>S</C> and
 <A>digraph</A> belongs to the category <Ref Filt="IsCayleyDigraph"/>, then
 <C>SemigroupOfCayleyDigraph</C> returns <C>S</C>.
 

 See also <Ref Attr="GeneratorsOfCayleyDigraph"/>.
 <Example><![CDATA[
gap> G := DihedralGroup(IsPermGroup, 8);
Group([ (1,2,3,4), (2,4) ])
gap> digraph := CayleyDigraph(G);
<immutable digraph with 8 vertices, 16 edges>
gap> GroupOfCayleyDigraph(digraph) = G;
true
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="GeneratorsOfCayleyDigraph">
<ManSection>
 <Attr Name="GeneratorsOfCayleyDigraph" Arg="digraph"/>
 <Returns>A list of generators.</Returns>
 <Description>
 If <A>digraph</A> is an immutable Cayley graph of a group or semigroup with
 respect to a set of generators <C>gens</C> and <A>digraph</A> belongs to
 the category <Ref Filt="IsCayleyDigraph"/>, then
 <C>GeneratorsOfCayleyDigraph</C> return the list of generators <C>gens</C>
 over which <A>digraph</A> is defined.
 

 See also <Ref Attr="GroupOfCayleyDigraph"/>
 or <Ref Attr="SemigroupOfCayleyDigraph"/>.
 <Example><![CDATA[
gap> G := DihedralGroup(IsPermGroup, 8);
Group([ (1,2,3,4), (2,4) ])
gap> digraph := CayleyDigraph(G);
<immutable digraph with 8 vertices, 16 edges>
gap> GeneratorsOfCayleyDigraph(digraph) = GeneratorsOfGroup(G);
true
gap> digraph := CayleyDigraph(G, [()]);
<immutable digraph with 8 vertices, 8 edges>
gap> GeneratorsOfCayleyDigraph(digraph) = [()];
true]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="EdgeOrbitsDigraph">
<ManSection>
 <Oper Name="EdgeOrbitsDigraph" Arg="G, edges[, n]"/>
 <Returns>
 An immutable digraph.
 </Returns>
 <Description>
 If <A>G</A> is a permutation group, <A>edges</A> is an edge or list of
 edges, and <A>n</A> is a non-negative integer such that <A>G</A> fixes
 <C>[1 .. <A>n</A>]</C> setwise, then
 this operation returns an immutable digraph with <A>n</A> vertices and the union
 of the orbits of the edges in <A> edges </A> under the action of the
 permutation group <A>G</A>. An edge in this context is simply a pair of
 positive integers.
 

 If the optional third argument <A>n</A> is not present, then the largest
 moved point of the permutation group <A>G</A> is used by default.

 <Example><![CDATA[
gap> digraph := EdgeOrbitsDigraph(Group((1, 3), (1, 2)(3, 4)),
> [[1, 2], [4, 5]], 5);
<immutable digraph with 5 vertices, 12 edges>
gap> OutNeighbours(digraph);
[ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4, 5 ], [ 1, 3, 5 ], [ ] ]
gap> RepresentativeOutNeighbours(digraph);
[ [ 2, 4, 5 ], [ ] ]
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphAddEdgeOrbit">
<ManSection>
 <Oper Name="DigraphAddEdgeOrbit" Arg="digraph, edge"/>
 <Returns>
 A new digraph.
 </Returns>
 <Description>
 This operation returns a new digraph with the same vertices and edges as
 <A>digraph</A> and with additional edges consisting of the orbit of the
 edge <A>edge</A> under the action of the <Ref Attr="DigraphGroup"/> of
 <A>digraph</A>. If <A>edge</A> is already an edge in <A>digraph</A>, then
 <A>digraph</A> is returned unchanged. The argument <A>digraph</A> must be
 an immutable digraph.
 

 An edge is simply a pair of vertices of <A>digraph</A>.

 <Example><![CDATA[
gap> gr1 := CayleyDigraph(DihedralGroup(8));
<immutable digraph with 8 vertices, 24 edges>
gap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);
<immutable digraph with 8 vertices, 32 edges>
gap> DigraphEdges(gr1);
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 5 ], [ 2, 6 ],
 [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ],
 [ 5, 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ],
 [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 5 ] ]
gap> DigraphEdges(gr2);
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 5 ],
 [ 2, 6 ], [ 2, 7 ], [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 3, 6 ],
 [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 7 ], [ 5, 6 ],
 [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], [ 6, 3 ],
 [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 7, 2 ], [ 8, 3 ], [ 8, 2 ],
 [ 8, 5 ], [ 8, 1 ] ]
gap> gr3 := DigraphRemoveEdgeOrbit(gr2, [1, 8]);
<immutable digraph with 8 vertices, 24 edges>
gap> gr3 = gr1;
true
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphRemoveEdgeOrbit">
<ManSection>
 <Oper Name="DigraphRemoveEdgeOrbit" Arg="digraph, edge"/>
 <Returns>
 A new digraph.
 </Returns>
 <Description>
 This operation returns a new digraph with the same vertices as
 <A>digraph</A> and with the orbit of the edge <A>edge</A> (under the action
 of the <Ref Attr="DigraphGroup"/> of <A>digraph</A>) removed.
 If <A>edge</A> is not an edge in <A>digraph</A>, then <A>digraph</A> is
 returned unchanged. The argument <A>digraph</A> must be an immutable
 digraph.
 

 An edge is simply a pair of vertices of <A>digraph</A>.
 <Example><![CDATA[
gap> gr1 := CayleyDigraph(DihedralGroup(8));
<immutable digraph with 8 vertices, 24 edges>
gap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);
<immutable digraph with 8 vertices, 32 edges>
gap> DigraphEdges(gr1);
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 5 ], [ 2, 6 ],
 [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ],
 [ 5, 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ],
 [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 5 ] ]
gap> DigraphEdges(gr2);
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 5 ],
 [ 2, 6 ], [ 2, 7 ], [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 3, 6 ],
 [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 7 ], [ 5, 6 ],
 [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], [ 6, 3 ],
 [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 7, 2 ], [ 8, 3 ], [ 8, 2 ],
 [ 8, 5 ], [ 8, 1 ] ]
gap> gr3 := DigraphRemoveEdgeOrbit(gr2, [1, 8]);
<immutable digraph with 8 vertices, 24 edges>
gap> gr3 = gr1;
true
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

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