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# 1 Grape

### Sections

1. Installing the GRAPE Package
2. Loading GRAPE
3. The structure of a graph in GRAPE
4. Examples of the use of GRAPE

This manual describes the GRAPE (Version 4.3) package for computing with graphs and groups.

GRAPE is primarily designed for the construction and analysis of finite graphs related to groups, designs, and geometries. Special emphasis is placed on the determination of regularity properties and subgraph structure. The GRAPE philosophy is that a graph gamma always comes together with a known subgroup G of the automorphism group of gamma, and that G is used to reduce the storage and CPU-time requirements for calculations with gamma (see Soi93 and Soi04). Of course G may be the trivial group, and in this case GRAPE algorithms may perform more slowly than strictly combinatorial algorithms (although this degradation in performance is hopefully never more than a fixed constant factor).

Most GRAPE functions are written entirely in the GAP language. However, the GRAPE functions `AutomorphismGroup`, `AutGroupGraph`, `IsIsomorphicGraph`, `GraphIsomorphismClassRepresentatives`, `GraphIsomorphism` and `PartialLinearSpaces` make direct or indirect use of B.D. McKay's nauty (Version 2.2 final) package Nau90, via a GRAPE interface. These functions can only be used on a fully installed version of GRAPE in a UNIX environment. Installation of GRAPE is described in its `README` file and in its manual section Installing the GRAPE Package.

Except for the nauty package included with GRAPE, and the function `SmallestImageSet` by Steve Linton, the GRAPE package was designed and written by Leonard H. Soicher, School of Mathematical Sciences, Queen Mary, University of London.

If you use GRAPE to solve a problem then please send a short email about it to L.H.Soicher@qmul.ac.uk, and reference the GRAPE package as follows:

L.H. Soicher, The GRAPE package for GAP, Version 4.3, 2006, http://www.maths.qmul.ac.uk/~leonard/grape/.

If your work made use a function depending on the nauty package then you should also reference nauty Nau90.

The development of GRAPE was partially supported by a European Union HCM grant in ``Computational Group Theory''.

## 1.1 Installing the GRAPE Package

To install GRAPE 4.3 (after installing GAP), first obtain the GRAPE archive file `grape4r3.tar.gz`, available at http://www.maths.qmul.ac.uk/~leonard/grape/grape4r3.tar.gz and then copy this archive file into the `pkg` directory of the GAP root directory. Actually, it is possible to have several GAP root directories (see GAP Root Directory), and so it is easy to install GRAPE locally even if you have no permission to add files to the main GAP installation. Now go to the appropriate `pkg` directory containing `grape4r3.tar.gz`, and then run

    gunzip grape4r3.tar.gz
    tar -xf grape4r3.tar

to unpack GRAPE.

After unpacking GRAPE, the GAP-only part of GRAPE is installed. The parts of GRAPE depending on B. D. Mckay's nauty package (for computing automorphism groups and testing graph isomorphism) are only available in a UNIX environment, where you should proceed as follows:

Go to the newly created `grape` directory (which we shall refer to as the home directory of GRAPE) and run `/bin/sh configure` path``, where path is the path to the home directory of the GAP distribution. So for example, if you install GRAPE in the `pkg` directory of the GAP home directory, run

    /bin/sh configure ../..

This will fetch the name of the system architecture on which GAP has been compiled, and create a `Makefile`. Now run

    make

to create the binaries and to put them in the appropriate place. This completes the installation of GRAPE for a single architecture. If GAP is installed on different architectures on a common file system, this configuration process will only work for the last architecture for which GAP was compiled. Therefore, you should always follow the above procedure to install the GRAPE binaries immediately after compiling GAP on a given architecture. However, if you want to add GRAPE later, you can just run `/bin/sh configure` again in the main GAP directory for the architecture before installing the GRAPE binaries for that architecture.

You should now test GRAPE and the interface to nauty on each architecture on which you have installed GRAPE. Start up GAP and at the prompt type

    LoadPackage( "grape" );

On-line documentation for GRAPE should be available by typing

    ?GRAPE

The command

    IsIsomorphicGraph( JohnsonGraph(7,3), JohnsonGraph(7,4) );

should return `true`, and

    Size( AutGroupGraph( JohnsonGraph(4,2) ) );

should be `48`.

Both dvi and pdf versions of the GRAPE manual are available (as `manual.dvi` and `manual.pdf` respectively) in the `doc` directory of the home directory of GRAPE.

If you install GRAPE, then please tell L.H.Soicher@qmul.ac.uk, where you should also send any comments or bug reports.

## 1.2 Loading GRAPE

Before using GRAPE you must load the package within GAP by calling the statement

    gap> LoadPackage("grape");

    Loading  GRAPE 4.3  (GRaph Algorithms using PErmutation groups),
    by L.H.Soicher@qmul.ac.uk.

    true

## 1.3 The structure of a graph in GRAPE

In general GRAPE deals with finite directed graphs which may have loops but have no multiple edges. However, many GRAPE functions only work for simple graphs (i.e. no loops, and whenever [x,y] is an edge then so is [y,x]), but these functions will check if an input graph is simple.

In GRAPE, a graph gamma is stored as a record, with mandatory components `isGraph`, `order`, `group`, `schreierVector`, `representatives`, and `adjacencies`. Usually, the user need not be aware of this record structure, and is strongly advised only to use GRAPE functions to construct and modify graphs.

The `order` component contains the number of vertices of gamma. The vertices of gamma are always 1,2,...,``gamma`.order`, but they may also be given names, either by a user (using `AssignVertexNames`) or by a function constructing a graph (e.g. `InducedSubgraph`, `BipartiteDouble`, `QuotientGraph`). The `names` component, if present, records these names, with ``gamma`.names[`i`]` the name of vertex i. If the `names` component is not present (the user may, for example, choose to unbind it), then the names are taken to be 1,2,...,``gamma`.order`. The `group` component records the GAP permutation group associated with gamma (this group must be a subgroup of the automorphism group of gamma). The `representatives` component records a set of orbit representatives for the action of ``gamma`.group` on the vertices of gamma, with ``gamma`.adjacencies[`i`]` being the set of vertices adjacent to ``gamma`.representatives[`i`]`. The `group` and `schreierVector` components are used to compute the adjacency-set of an arbitrary vertex of gamma (this is done by the function `Adjacency`).

The only mandatory component which may change once a graph is initially constructed is `adjacencies` (when an edge-orbit of ``gamma`.group` is added to, or removed from, gamma). A graph record may also have some of the optional components `isSimple`, `autGroup`, and `canonicalLabelling`, which record information about that graph.

Note All global variables used by GRAPE start with `GRAPE_`.

## 1.4 Examples of the use of GRAPE

We give here a simple example to illustrate the use of GRAPE. All functions used are described in detail in this manual. More sophisticated examples of the use of GRAPE can be found in chapter Partial Linear Spaces, and also in the references Cam99, CSS99, HL99 and Soi06.

In the example here, we construct the Petersen graph P, and its edge graph (also called line graph) EP. We compute the global parameters of EP, and so verify that EP is distance-regular (see BCN89).

    gap> LoadPackage("grape");

    Loading  GRAPE 4.3  (GRaph Algorithms using PErmutation groups),
    by L.H.Soicher@qmul.ac.uk.

    true
    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, 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 ] ] )
    gap> Diameter(P);
    2
    gap> Girth(P);
    5
    gap> EP := EdgeGraph(P);
    rec( isGraph := true, order := 15,
      group := Group([ ( 1, 4, 7, 2, 5)( 3, 6, 8, 9,12)(10,13,14,15,11),
          ( 4, 9)( 5,11)( 6,10)( 7, 8)(12,15)(13,14) ]),
      schreierVector := [ -1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2 ],
      adjacencies := [ [ 2, 3, 7, 8 ] ], representatives := [ 1 ],
      isSimple := true,
      names := [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 2 ], [ 4, 5 ] ],
          [ [ 1, 2 ], [ 3, 5 ] ], [ [ 2, 3 ], [ 4, 5 ] ], [ [ 2, 3 ], [ 1, 5 ] ],
          [ [ 2, 3 ], [ 1, 4 ] ], [ [ 3, 4 ], [ 1, 5 ] ], [ [ 3, 4 ], [ 2, 5 ] ],
          [ [ 1, 3 ], [ 4, 5 ] ], [ [ 1, 3 ], [ 2, 4 ] ], [ [ 1, 3 ], [ 2, 5 ] ],
          [ [ 2, 4 ], [ 1, 5 ] ], [ [ 2, 4 ], [ 3, 5 ] ], [ [ 3, 5 ], [ 1, 4 ] ],
          [ [ 1, 4 ], [ 2, 5 ] ] ] )
    gap> GlobalParameters(EP);
    [ [ 0, 0, 4 ], [ 1, 1, 2 ], [ 1, 2, 1 ], [ 4, 0, 0 ] ]

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