This manual describes the GRAPE (Version 4.9.0) 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).
Certain GRAPE functions make direct or indirect use of the nauty MP14 or bliss JK07 packages, for computing automorphism groups of graphs and testing graph isomorphism. Such functions can only be used on a fully installed version of GRAPE. Installation of GRAPE is described in this chapter of the manual.
Except for the nauty package of B. D. McKay included with GRAPE,
the function SmallestImageSet by Steve Linton, the nauty interface
by Alexander Hulpke, and the initial bliss interface by Jerry James,
the GRAPE package was designed and written by Leonard H. Soicher,
School of Mathematical Sciences, Queen Mary University of London.
Except for the included nauty package, GRAPE is licensed under the
terms of the GNU General Public License as published by the Free Software
Foundation; either version 2 of the License, or (at your option) any later
version. For details, see https://www.gnu.org/licenses/gpl.html.
Further licensing and copyright information for GRAPE is contained
in its README.md file.
If you use GRAPE in a published work, then please reference the package as follows:
L.H. Soicher, The GRAPE package for GAP, Version 4.9.0, 2022, https://gap-packages.github.io/grape.
For questions, remarks, suggestions, and issues, please use the issue tracker at https://github.com/gap-packages/grape/issues.
The development of GRAPE was partially supported by a European Union HCM grant in ``Computational Group Theory'', and more recently by EPSRC grant EP/M022641/1 (CoDiMa: a Collaborative Computational Project in the area of Computational Discrete Mathematics).
The official GAP Windows distribution includes the GRAPE package fully installed. Thus, GRAPE normally requires no further installation for Windows users of GAP. What follows is for Unix users of GRAPE.
You do not need to download and unpack an archive for GRAPE unless you
want to install the package separately from your main GAP installation
or are installing an upgrade of GRAPE to an existing installation
of GAP (see the main GAP reference section Installing a GAP Package). If you do need to download GRAPE, you can find the most
recent .tar.gz archive at https://gap-packages.github.io/grape.
The archive file should be downloaded and unpacked in the pkg
subdirectory of an appropriate GAP root directory (see the main GAP
reference section GAP Root Directories).
If your GRAPE installation does not include a pre-compiled binary of
the nauty/dreadnaut programs included with GRAPE and you do not want
to use an already installed version of nauty or bliss, you will
need to perform compilation of the nauty/dreadnaut programs included with
GRAPE, and to do this in a Unix environment, you should proceed as
follows. After installing GAP, go to the GRAPE home directory
(usually the directory pkg/grape of the GAP home directory),
and run ./configure path, where path is the path of the GAP
home directory. So for example, if you install GRAPE in the pkg
directory of the GAP home directory, run
./configure ../..This will fetch the name of the architecture for which GAP has been most recently configured, and create a
Makefile. Now run
maketo create the nauty/dreadnaut binary and to put it in the appropriate place. This configuration/make process for GRAPE only works for the last architecture for which GAP was configured. Therefore, you should always follow the above procedure to install the nauty/dreadnaut binary immediately after compiling GAP for a given configuration, say for a different architecture on a common file system. However, if you want to add GRAPE later, you can just run
./configure again in the GAP home directory for the architecture,
before performing the GRAPE configure/make process to install the
nauty/dreadnaut binary for that architecture.
To use GRAPE with a separately installed version of nauty or
bliss you should proceed as follows. Please note that the nauty
interface for GRAPE has only been extensively tested with the
included versions of nauty, and the bliss interface has only
been tested with Version 0.73 of bliss. To use a separately
installed version of nauty, type the following commands in GAP, or
place these commands in your gaprc file (see The gaprc file), where
dreadnaut_or_dreadnautB_executable should be the name of your
dreadnaut or dreadnautB executable file:
LoadPackage("grape");
GRAPE_NAUTY := true;
GRAPE_DREADNAUT_EXE := "dreadnaut_or_dreadnautB_executable";
To use a separately installed version of bliss instead of nauty,
type the following commands in GAP, or place these commands in your
gaprc file (see The gaprc file), where bliss_executable should be
the name of your bliss executable file:
LoadPackage("grape");
GRAPE_NAUTY := false;
GRAPE_BLISS_EXE := "bliss_executable";
For example, if the bliss executable is /usr/local/bin/bliss, then type:
LoadPackage("grape");
GRAPE_NAUTY := false;
GRAPE_BLISS_EXE := "/usr/local/bin/bliss";
You should now test GRAPE and the interface to nauty or bliss 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
?GRAPEThen run some tests by typing:
Test(Filename(DirectoriesPackageLibrary("grape","tst"),"testall.tst"));
This should return the value true.
A pdf version of the GRAPE manual is available as manual.pdf in the
doc directory of the home directory of GRAPE.
Before using GRAPE you must load the package within GAP by typing:
LoadPackage("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, maximumClique,
minimumVertexColouring, and canonicalLabelling, which record
information about that graph.
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");
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 ] ]
grape manual