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<h1>1 Grape</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP001.htm#SECT001">Installing the GRAPE Package</a>
<li> <A HREF="CHAP001.htm#SECT002">Loading GRAPE</a>
<li> <A HREF="CHAP001.htm#SECT003">The structure of a graph in GRAPE</a>
<li> <A HREF="CHAP001.htm#SECT004">Examples of the use of GRAPE</a>
</ol>

This manual describes the GRAPE (Version 4.9.3) 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 <var>gamma</var>
always comes together with a known subgroup <var>G</var> of the automorphism
group of <var>gamma</var>, and that <var>G</var> is used to reduce the storage and
CPU-time requirements for calculations with <var>gamma</var> (see
<a href="biblio.htm#Soi93"><cite>Soi93</cite></a> and <a href="biblio.htm#Soi04"><cite>Soi04</cite></a>). Of course <var>G</var> 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
<a href="biblio.htm#MP14"><cite>MP14</cite></a> or bliss <a href="biblio.htm#JK07"><cite>JK07</cite></a> 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 <code>SmallestImageSet</code> 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 <a href="https://www.gnu.org/licenses/gpl.html">https://www.gnu.org/licenses/gpl.html</a>.
Further licensing and copyright information for GRAPE is contained
in its <code>README.md</code> 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 4.9.3, 2025,
<a href="https://gap-packages.github.io/grape">https://gap-packages.github.io/grape</a>.

For questions, remarks, suggestions, and issues, please use the issue
tracker at <a href="https://github.com/gap-packages/grape/issues">https://github.com/gap-packages/grape/issues</a>.

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).


<h2><a name="SECT001">1.1 Installing the GRAPE Package</a></h2>

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 <a href="../../../doc/ref/chap76.html#X82473E4B8756C6CD">Installing a GAP Package</a>). If you do need to download GRAPE, you can find the most
recent <code>.tar.gz</code> archive at <a href="https://gap-packages.github.io/grape">https://gap-packages.github.io/grape</a>.
The archive file should be downloaded and unpacked in the <code>pkg</code>
subdirectory of an appropriate GAP root directory (see the main GAP
reference section <a href="../../../doc/ref/chap9.html#X7A4973627A5DB27D">GAP Root Directories</a>).

If your GRAPE installation does not include a 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 <code>pkg/grape</code> of the GAP home directory),
and run <code>./configure </code><var>path</var><code></code>, where <var>path</var> is the path of the GAP
home directory. So for example, if you install GRAPE in the <code>pkg</code>
directory of the GAP home directory, run
<pre>
./configure ../..
</pre>
Then run
<pre>
make
</pre>
to complete the installation of GRAPE.

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 <code>gaprc</code> file (see <a href="../../../doc/ref/chap3.html#X84D4CF587D437C00">The gaprc file</a>), where
<code>dreadnaut_or_dreadnautB_executable</code> should be the name of your
dreadnaut or dreadnautB executable file:
<pre>
LoadPackage("grape");
GRAPE_NAUTY := true;
GRAPE_DREADNAUT_EXE := "dreadnaut_or_dreadnautB_executable";
</pre>
To use a separately installed version of bliss instead of nauty,
type the following commands in GAP, or place these commands in your
<code>gaprc</code> file (see <a href="../../../doc/ref/chap3.html#X84D4CF587D437C00">The gaprc file</a>), where <code>bliss_executable</code> should be
the name of your bliss executable file:
<pre>
LoadPackage("grape");
GRAPE_NAUTY := false;
GRAPE_BLISS_EXE := "bliss_executable";
</pre>
For example, if the bliss executable is <code>/usr/local/bin/bliss</code>, then type:
<pre>
LoadPackage("grape");
GRAPE_NAUTY := false;
GRAPE_BLISS_EXE := "/usr/local/bin/bliss";
</pre>

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
<pre>
LoadPackage( "grape" );
</pre>
On-line documentation for GRAPE should be available by typing
<pre>
?GRAPE
</pre>
Then run some tests by typing:
<pre>
Test(Filename(DirectoriesPackageLibrary("grape","tst"),"testall.tst"));
</pre>
This should return the value <code>true</code>.

A pdf version of the GRAPE manual is available as <code>manual.pdf</code> in the
<code>doc</code> directory of the home directory of GRAPE.


<h2><a name="SECT002">1.2 Loading GRAPE</a></h2>

Before using GRAPE you must load the package within GAP by typing:

<pre>
LoadPackage("grape");
</pre>


<h2><a name="SECT003">1.3 The structure of a graph in GRAPE</a></h2>

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 <var>[x,y]</var> is an
edge then so is <var>[y,x]</var>), but these functions will check if an input
graph is simple.

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

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


<h2><a name="SECT004">1.4 Examples of the use of GRAPE</a></h2>

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 <a href="CHAP009.htm#SSEC001.1">Partial Linear Spaces</a>, and also in the references <a href="biblio.htm#Cam99"><cite>Cam99</cite></a>,
<a href="biblio.htm#CSS99"><cite>CSS99</cite></a>, <a href="biblio.htm#HL99"><cite>HL99</cite></a>, <a href="biblio.htm#Soi06"><cite>Soi06</cite></a> and <a href="biblio.htm#Soi24b"><cite>Soi24b</cite></a>.

In the example here, we construct the Petersen graph <var>P</var>, and its edge
graph (also called line graph) <var>EP</var>. We compute the global parameters
of <var>EP</var>, and so verify that <var>EP</var> is distance-regular (see <a href="biblio.htm#BCN89"><cite>BCN89</cite></a>).

<pre>
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 ] ]
</pre>

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