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\title{\LaTeX\ facilities, II}
\author{Josef Lauri}
\date{ }


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\maketitle

\begin{abstract}
This short document is intended to show those students needing to
use \LaTeX\ a simple example of how to include citations using
using \BibTeX. Extensive information about using \BibTeX\ can be
found at www.bibtex.org. The inclusion of a table of contents is
also illustrated. The original \LaTeX\ file and the .bib file are
also available. Note that the .bib file contains more entries than
shown here. Only the cited references are extracted and shown in
the final document.
\end{abstract}

\tableofcontents

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\section{Graphs and digraphs}


A number of definitions on graphs and digraphs will be given as
they are required. However, several standard graph theoretic terms
will be used but not defined in these chapters; these can be found
in any of the references \cite{west96} or \cite{wilson97}.

\subsection{An example of a subsection}

A very short subsection!

\section{Groups}

Counting nonisomorphic graphs involves consideration of group
symmetries. For more on this the reader is referred to
\cite{harary&pal73}.


\subsection{And here is another subsection}

The following theorem of Whitney \cite{whitney32} says that these
are essentially the only cases when edge-isomorphisms that are not
induced by isomorphisms can arise. We give the statement of the
theorem without proof, which, although not deep or difficult,
would lengthen this introductory chapter without adding
significant new insights.

\subsection{And another}

Of course, there are several terms in the previous sentence that
need exact definitions, but we shall here take an intuitive
approach and refer the reader to \cite{bovet&cre94} or
\cite{garey&joh79} for the exact details on computational
complexity.

\section{The next section}

An important computer algebra package, which is also freely
available, is the system {\bf\sf GAP} \cite{gap99}. This package
performs very sophisticated routines in discrete abstract algebra,
in particular routines on permutation groups. It incorporates a
number of extensions, one of which, {\bf\sf GRAPE}
\cite{soicher93}, deals specifically with graphs, including their
automorphisms and isomorphisms.

\subsection{More references}

Finally, it should be mentioned that it is generally accepted that
the best package to tackle graph isomorphisms is {\it nauty}
\cite{mckay90}, developed by Brendan McKay. In fact, the system
{\bf\sf GRAPE} invokes {\it nauty} when computing automorphisms or
isomorphisms.

\section{Notes and guide to references}

One of the standard texts on graph theory has, for many years,
been \cite{harary69}. More recent books that give an excellent
coverage of the subject are
\cite{bollobas79,diestel97,west96,wilson97}. The last reference is
a short introduction that is quite sufficient background for this
book. Biggs' book \cite{biggs93} is the standard text on algebraic
graph theory, but the more recent \cite{godsil&roy01} is also an
excellent and up-to-date textbook on the subject. The book
\cite{hahn&sab97} contains a number of recent and specialised
survey papers on various aspects of algebraic graph theory,
particularly those dealing with graph symmetries. A proof of
Whitney's Theorem can be found in \cite{behzad&79}.

We shall only be needing the most elementary notions of group
theory. The text \cite{lederman&wei96} gives ample coverage for
our purposes, while \cite{rotman95} provides a more complete
treatment. Two excellent books devoted entirely to permutation
groups are \cite{cameron99,dixon&mor96}. Most of the results and
definitions on permutation groups that we have given here and
others that we shall be needing can be found in the first few
chapters of these two books.

For a full discussion of the terms on computational complexity
that were introduced above rather intuitively, the reader is
referred to the standard textbook \cite{garey&joh79} or the more
recent \cite{bovet&cre94}. The book \cite{kobler&93} and the
references that it cites are suggested for those who are
interested in the computational complexity of the graph
isomorphism problem. Those who are particularly interested in some
of the powerful algebraic techniques used to tackle this problem
should look at the papers \cite{hoffman82,luks82}. For practical
computations on a computer with permutation groups and graph
automorphisms and isomorphisms in particular, the system
\cite{g&g}, mentioned above, or the systems
\cite{mckay90,soicher93,gap99} are recommended.

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