[GRAPE] 2 Functions to construct and modify graphs

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# 2 Functions to construct and modify graphs

### Sections

1. Graph
2. EdgeOrbitsGraph
3. NullGraph
4. CompleteGraph
5. JohnsonGraph
6. CayleyGraph
7. AddEdgeOrbit
8. RemoveEdgeOrbit
9. AssignVertexNames

This chapter describes the functions used to construct and modify graphs.

## 2.1 Graph

- `Graph(` G`,` L`,` act`,` rel`)`
- `Graph(` G`,` L`,` act`,` rel`,` invt`)`

  This is the most general and useful way of constructing a graph in GRAPE.

  First suppose that the optional boolean parameter invt is unbound or has value `false`. Then L should be a list of elements of a set S on which the group G acts, with the action given by the function act. The parameter rel should be a boolean function defining a G-invariant relation on S (so that for g in G, x,y in S, rel (x,y) if and only if rel (act (x,g),act (y,g))). Then the function `Graph` returns a graph gamma which has as vertex-names (an immutable copy of)

  centerline`Concatenation( Orbits(` G`,` L`,` act`) )`

  (the concatenation of the distinct orbits of the elements in L under G), and for vertices v,w of gamma, [v,w] is an edge if and only if

  centerline``rel`( VertexName(` gamma`,` v`), VertexName(` gamma`,` w`) )`.

  Now if the parameter invt exists and has value `true`, then it is assumed that L is invariant under G with respect to action act. Then the function `Graph` behaves as above, except that the vertex-names of gamma become (an immutable copy of) L.

  The group associated with the graph gamma returned is the image of G acting via act on ``gamma`.names`.

  For example, suppose you have an n by n adjacency matrix A for a graph X, so that the vertex-set of X is {1,¼, n}, and [i,j] is an edge of X if and only if A[i][j]=1. Suppose also that G £ \Aut (X) (G may be trivial). Then you can make a GRAPE graph isomorphic to X via `Graph( G, [1..n], OnPoints, function(x,y) return A[x][y]=1; end, true );`

      gap> A := [[0,1,0],[1,0,0],[0,0,1]];
      [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ]
      gap> G := Group( (1,2) );
      Group([ (1,2) ])
      gap> Graph( G, [1..3], OnPoints,
      >        function(x,y) return A[x][y]=1; end,
      >        true );
      rec(
        isGraph := true,
        order := 3,
        group := Group( [ (1,2) ] ),
        schreierVector := [ -1, 1, -2 ],
        adjacencies := [ [ 2 ], [ 3 ] ],
        representatives := [ 1, 3 ],
        names := [ 1, 2, 3 ] )

  We now use `Graph` to construct the Petersen graph.

      gap> Petersen := Graph( SymmetricGroup(5), [[1,2]], OnSets,
      >                    function(x,y) return Intersection(x,y)=[]; end );
      rec(
        isGraph := true,
        order := 10,
        group := Group( [ ( 1, 2, 3, 5, 7)( 4, 6, 8, 9,10), ( 2, 4)( 6, 9)( 7,10)
           ] ),
        schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ],
        adjacencies := [ [ 3, 5, 8 ] ],
        representatives := [ 1 ],
        names := [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], [ 2, 4 ],
            [ 1, 5 ], [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ] )

  ## 2.2 EdgeOrbitsGraph

- `EdgeOrbitsGraph(` G`,` edges`)`
- `EdgeOrbitsGraph(` G`,` edges`,` n`)`

  This is a common way of constructing a graph in GRAPE.

  This function returns the (directed) graph with vertex-set {1,¼, n }, edge-set Èe Î edges eG , and associated (permutation) group G, which must act naturally on {1,¼,n }. The parameter edges should be a list of edges (lists of length 2 of vertices), although a singleton edge will be understood as an edge-list of length 1. The parameter n may be omitted, in which case n is taken to be the largest point moved by G.

  Note that G may be the trivial permutation group (`Group( () )` in GAP notation), in which case the (directed) edges of gamma are simply those in the list edges.

      gap> EdgeOrbitsGraph( Group((1,3),(1,2)(3,4)), [[1,2],[4,5]], 5 );
      rec(
        isGraph := true,
        order := 5,
        group := Group( [ (1,3), (1,2)(3,4) ] ),
        schreierVector := [ -1, 2, 1, 2, -2 ],
        adjacencies := [ [ 2, 4, 5 ], [  ] ],
        representatives := [ 1, 5 ],
        isSimple := false )

  ## 2.3 NullGraph

- `NullGraph(` G`)`
- `NullGraph(` G`,` n`)`

  This function returns the null graph (graph with no edges) with vertex-set {1,¼,n }, and associated (permutation) group G. The parameter n may be omitted, in which case n is taken to be the largest point moved by G.

  See also IsNullGraph.

      gap> NullGraph( Group( (1,2,3) ), 4 );
      rec(
        isGraph := true,
        order := 4,
        group := Group( [ (1,2,3) ] ),
        schreierVector := [ -1, 1, 1, -2 ],
        adjacencies := [ [  ], [  ] ],
        representatives := [ 1, 4 ],
        isSimple := true )

  ## 2.4 CompleteGraph

- `CompleteGraph(` G`)`
- `CompleteGraph(` G`,` n`)`
- `CompleteGraph(` G`,` n`,` mustloops`)`

  This function returns the complete graph with vertex-set {1,¼,n } and associated (permutation) group G. The parameter n may be omitted, in which case n is taken to be the largest point moved by G. The optional boolean parameter mustloops determines whether the complete graph has all loops present or no loops (default: `false` (no loops)).

  See also IsCompleteGraph.

      gap> CompleteGraph( Group( (1,2,3), (1,2) ) );
      rec(
        isGraph := true,
        order := 3,
        group := Group( [ (1,2,3), (1,2) ] ),
        schreierVector := [ -1, 1, 1 ],
        adjacencies := [ [ 2, 3 ] ],
        representatives := [ 1 ],
        isSimple := true )

  ## 2.5 JohnsonGraph

- `JohnsonGraph(` n`,` e`)`

  Let n and e be integers, with n ³ e ³ 0. Then this function returns a graph gamma isomorphic to the Johnson graph J(n ,e ). The vertices (actually the vertex-names) of gamma are the e-subsets of {1,¼, n }, with x joined to y if and only if |x Çy| = e -1. The group associated with gamma is the image of the symmetric group Sn acting on the e-subsets of {1,¼,n }.

      gap> JohnsonGraph(5,3);
      rec(
        isGraph := true,
        order := 10,
        group := Group( [ ( 1, 7,10, 6, 3)( 2, 8, 4, 9, 5), ( 4, 7)( 5, 8)( 6, 9)
           ] ),
        schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 2, 1, 1 ],
        adjacencies := [ [ 2, 3, 4, 5, 7, 8 ] ],
        representatives := [ 1 ],
        names := [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 2, 5 ], [ 1, 3, 4 ], [ 1, 3, 5 ],
            [ 1, 4, 5 ], [ 2, 3, 4 ], [ 2, 3, 5 ], [ 2, 4, 5 ], [ 3, 4, 5 ] ],
        isSimple := true )

  ## 2.6 CayleyGraph

- `CayleyGraph(` G`)`
- `CayleyGraph(` G`,` gens`)`
- `CayleyGraph(` G`,` gens`,` undirected`)`

  Given a group G and a generating list gens for G, `CayleyGraph(` G`,` gens`)` returns a Cayley graph for G with respect to gens. The generating list gens is optional, and if omitted, then gens is taken to be `GeneratorsOfGroup(` G`)`. The boolean argument undirected is also optional, and if undirected=`true` (the default), then the returned graph is undirected (as if gens was closed under inversion, whether or not it is).

  The Cayley graph caygraph which is returned is defined as follows: the vertices (actually the vertex-names) of caygraph are the elements of G; if undirected=`true` (the default) then vertices x,y are joined by an edge if and only if there is a g in the list gens with y=gx or y=g-1x; if undirected=`false` then [x,y] is an edge if and only if there is a g in gens with y=gx.

  The permutation group ``caygraph`.group` associated with caygraph is the image of G acting in its right regular representation.

  Note It is not checked whether G is actually generated by gens. However, even if G is not generated by gens, the function still works as described above (as long as gens is contained in G), but returns a ``Cayley graph'' which is not connected.

      gap> C:=CayleyGraph(SymmetricGroup(4),[(1,2),(2,3),(3,4)]);
      rec(
        isGraph := true,
        order := 24,
        group :=
         Group( [ ( 1,10,17,19)( 2, 9,18,20)( 3,12,14,21)( 4,11,13,22)( 5, 7,16,23)
              ( 6, 8,15,24), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,15)
              (14,16)(17,18)(19,21)(20,22)(23,24) ] ),
        schreierVector := [ -1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2,
            1, 1, 2, 2, 1, 2 ],
        adjacencies := [ [ 2, 3, 7 ] ],
        representatives := [ 1 ],
        names := [ (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4),
            (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4), (1,3,2), (1,3,4,2), (1,3),
            (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4),
            (1,4,2,3), (1,4)(2,3) ],
        isSimple := true )
      gap> Girth(C);
      4
      gap> Diameter(C);
      6

  ## 2.7 AddEdgeOrbit

- `AddEdgeOrbit(` gamma`,` e`)`
- `AddEdgeOrbit(` gamma`,` e`,` H`)`

  This procedure adds the orbit of e under ``gamma`.group` to the edge-set of the graph gamma. The parameter e must be a sequence of length 2 of vertices of gamma. If the optional third parameter H is given then it is assumed that ``e`[2]` has the same orbit under H as it does under the stabilizer in ``gamma`.group` of ``e`[1]`, and this knowledge can speed up the procedure.

  Note that if ``gamma`.group` is trivial then this procedure simply adds the single (directed) edge e to gamma.

  See also RemoveEdgeOrbit.

      gap> gamma := NullGraph( Group( (1,3), (1,2)(3,4) ) );;
      gap> AddEdgeOrbit( gamma, [4,3] );
      gap> gamma;
      rec(
        isGraph := true,
        order := 4,
        group := Group( [ (1,3), (1,2)(3,4) ] ),
        schreierVector := [ -1, 2, 1, 2 ],
        adjacencies := [ [ 2, 4 ] ],
        representatives := [ 1 ],
        isSimple := true )
      gap> GlobalParameters(gamma);
      [ [ 0, 0, 2 ], [ 1, 0, 1 ], [ 2, 0, 0 ] ]

  ## 2.8 RemoveEdgeOrbit

- `RemoveEdgeOrbit(` gamma`,` e`)`
- `RemoveEdgeOrbit(` gamma`,` e`,` H`)`

  This procedure removes the orbit of e under ``gamma`.group` from the edge-set of the graph gamma. The parameter e must be a sequence of length 2 of vertices of gamma, but if e is not an edge of gamma then this procedure has no effect. If the optional third parameter H is given then it is assumed that ``e`[2]` has the same orbit under H as it does under the stabilizer in ``gamma`.group` of ``e`[1]`, and this knowledge can speed up the procedure.

  See also AddEdgeOrbit.

      gap> gamma := CompleteGraph( Group( (1,3), (1,2)(3,4) ) );;
      gap> RemoveEdgeOrbit( gamma, [1,3] );
      gap> gamma;
      rec(
        isGraph := true,
        order := 4,
        group := Group( [ (1,3), (1,2)(3,4) ] ),
        schreierVector := [ -1, 2, 1, 2 ],
        adjacencies := [ [ 2, 4 ] ],
        representatives := [ 1 ],
        isSimple := true )
      gap> GlobalParameters(gamma);
      [ [ 0, 0, 2 ], [ 1, 0, 1 ], [ 2, 0, 0 ] ]

  ## 2.9 AssignVertexNames

- `AssignVertexNames(` gamma`,` names`)`

  This procedure allows the user to give new names for the vertices of gamma, by specifying a list names (of length ``gamma`.order`) of vertex-names for the vertices of gamma, such that ``names`[`i`]` contains the user's name for the i-th vertex of gamma.

  An immutable copy of names is assigned to ``gamma`.names`.

  See also VertexNames and VertexName.

      gap> gamma := NullGraph( Group(()), 3 );
      rec(
        isGraph := true,
        order := 3,
        group := Group( [ () ] ),
        schreierVector := [ -1, -2, -3 ],
        adjacencies := [ [  ], [  ], [  ] ],
        representatives := [ 1, 2, 3 ],
        isSimple := true )
      gap> AssignVertexNames( gamma, ["a","b","c"] );
      gap> gamma;
      rec(
        isGraph := true,
        order := 3,
        group := Group( [ () ] ),
        schreierVector := [ -1, -2, -3 ],
        adjacencies := [ [  ], [  ], [  ] ],
        representatives := [ 1, 2, 3 ],
        isSimple := true,
        names := [ "a", "b", "c" ] )

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