%revised version for E-JC
%'previous' version drgrevision

\documentclass[12pt]{article}
\usepackage{e-jc}

%\usepackage[backref=page]{hyperref}
\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue,backref=page]{hyperref}
\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:\linebreak[2]#1}}}

\usepackage{amssymb,amsmath,amsthm,enumerate,mathrsfs,url,graphicx,ifthen}
%\usepackage{color}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%BACK REFERENCES SET-UP

\renewcommand*{\backreflastsep}{, }%instead of separation of page numbers by "and"
\renewcommand*{\backreftwosep}{, }%idem
\renewcommand*{\backref}[1]{}
\renewcommand*{\backrefalt}[4]{%
%     \ifcase #1 (\textcolor{blue}{Not cited.})%
     \ifcase #1 (Not cited.)%
     \or        (Cited on p.~#2.)%
     \else      (Cited on pp.~#2.)%
     \fi}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\G}{\Gamma}
\newcommand{\gauss}[2]{\genfrac[]{0pt}{}{#1}{#2}}
\renewcommand{\binom}[2]{\genfrac(){0pt}{}{#1}{#2}}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conj}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{claim}{Claim}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{rk}{Remark}

\theoremstyle{remark}
\newtheorem{problem}{Problem}

\newtheorem*{biconj}{Bannai-Ito conjecture}


\newcommand{\R}{{\mathbb R}}

\newcommand{\al}{\ensuremath{\alpha}}
\newcommand{\be}{\ensuremath{\beta}}
\newcommand{\ld}{\ensuremath{\lambda}}
\newcommand{\Ga}{\ensuremath{\Gamma}}

\newcommand{\A}{\mathcal{A}}%dual polar graph
\newcommand{\B}{\mathcal{B}}%dual polar graph
\newcommand{\C}{\mathcal{C}}%dual polar graph
\newcommand{\D}{\mathcal{D}}%dual polar graph
\newcommand{\HH}{\mathcal{H}}%not used
\newcommand{\J}{\mathcal{J}}%not used

\newcommand{\AL}{\mathbb{A}}%adjacency algebra
\newcommand{\TT}{\mathbb{T}}%terwilliger algebra

\def\tr{\mathop{\rm tr }\nolimits}

\renewcommand{\cal}{\mathcal}

\newcommand{\ster}{\star}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{\bf Distance-regular graphs}
\author{Edwin R. van Dam\\
\small Department of Econometrics and O.R.\\[-0.8ex]
\small Tilburg University\\[-0.8ex]
%\small P.O. Box 90153, 5000 LE Tilburg\\[-0.8ex]
\small The Netherlands\\
\small\tt Edwin.vanDam@uvt.nl\\
\and
Jack H. Koolen\\
\small School of Mathematical Sciences\\[-0.8ex]
\small University of Science and Technology of China\\[-0.8ex]
\small and\\[-0.8ex]
\small Wu Wen-Tsun Key Laboratory of\\[-0.8ex]
\small Mathematics of CAS\\[-0.8ex]
\small Hefei, Anhui, 230026, China\\
\small\tt koolen@ustc.edu.cn\\
\and
Hajime Tanaka\\
\small Research Center for\\[-0.8ex]
\small Pure and Applied Mathematics\\[-0.8ex]
\small Graduate School of Information Sciences\\[-0.8ex]
\small Tohoku University\\[-0.8ex]
\small Sendai 980-8579, Japan\\
\small\tt htanaka@tohoku.ac.jp\\
}
\date{\dateline{Mar 31, 2016}{}\\
\small Mathematics Subject Classifications: 05E30, 05Cxx, 05Exx}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\date{\today. 2010 Mathematics Subject Classification: 05E30. This manuscript is unfinished.
%Comments on this version are very welcome.}
%\subjclass[2010]{05E30}

\begin{document}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}
\noindent This is a survey of distance-regular graphs. We present an
introduction to distance-regular graphs for the reader who is unfamiliar with
the subject, and then give an overview of some developments in the area of
distance-regular graphs since the monograph `BCN' [Brouwer, A.E., Cohen, A.M.,
Neumaier, A.,
    {\sl Distance-Regular Graphs}, Springer-Verlag, Berlin, 1989] was written.

\bigskip\noindent \textbf{Keywords:} Distance-regular graph; survey; association scheme; $P$-polynomial; $Q$-polynomial; geometric
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{tableofcontents}
\end{tableofcontents}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}\label{sec1:introduction}

\input{1_introduction.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{An introduction to distance-regular graphs}\label{sec2:basics}

\input{2_drgintroduction.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Examples}\label{sec3:classical}

\input{3_examples.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{More background}\label{sec:morebackground}

\input{4_morebackground.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{\texorpdfstring{$Q$-polynomial}{Q-polynomial} distance-regular graphs}\label{sec:Qpol}

\input{5_Qpolynomial.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Terwilliger algebra and combinatorics}\label{sec:Talg6}

\input{6_Talgebra.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Growth of intersection numbers and bounds on the diameter}\label{sec:diameterbounds}
%\section{Growth of intersection numbers and diameter bounds}\label{sec:diameterbounds}

\input{7_diameterbounds.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Bannai-Ito conjecture}\label{sec:BIconjecture}

\input{8_BIconjecture.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Geometric distance-regular graphs}\label{sec:Geometricdrg}

\input{9_geometric.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Spectral characterizations}\label{sec:spectralchar}

\input{10_spectralchar.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Subgraphs}\label{sec: subgraphs}

\input{11_subgraphs.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Completely regular codes}\label{sec:crc}

\input{12_crc.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{More combinatorial properties}\label{sec:morecombinatorial}

\input{13_morecomb.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Multiplicities}\label{sec:mult}

\input{14_multiplicities.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Applications}\label{sec:applications}

\input{15_applications.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Miscellaneous}\label{sec:misc}

\input{16_miscellaneous.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Tables}\label{sec:tables}

\input{17_tables.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Open problems and research directions}\label{sec12:Openproblems}

\input{18_openproblems.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip
\noindent {\bf Acknowledgements.}
The authors thank Eiichi Bannai, Andries Brouwer, Bart De
Bruyn, Miquel \`{A}ngel Fiol, Willem Haemers,
Akira Hiraki,
Tatsuro Ito, Sasha Juri\v{s}i\'{c}, Greg
Markowsky, Bill Martin, Akihiro Munemasa, Jongyook Park, Renata Sotirov,
Hiroshi Suzuki,
and Paul Terwilliger for inspiring
conversations on distance-regular graphs and this survey paper.

The authors also thank Sasha Gavrilyuk, Neil Gillespie, Akihide Hanaki, Arnold Neumaier, Nobuaki Obata, L\'{a}szl\'{o} Pyber, Achill Sch\"{u}rmann, Fr\'{e}d\'{e}ric
Vanhove, and a referee for comments on an earlier version of this paper and making some
valuable suggestions.

A significant part of the work was done while Jack Koolen was working at POSTECH.
Jack Koolen gratefully acknowledges the support of the `100 talents' program of the Chinese Academy of Sciences.
Research of Jack Koolen is partially supported by the National Natural Science Foundation of China (No. 11471009).
The work of Hajime Tanaka was supported by JSPS Grant-in-Aid
for Scientific Research No.~23740002 and No.~25400034.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\input{bibliography.txt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
