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

\subsection{The classical families with unbounded diameter}\label{sec:clasfamilies}

The {\em Johnson graph} $J(n,D)$ has as vertices the subsets of size $D$ of a set of size $n$. Two subsets are adjacent
if and only if they differ in precisely one element; cf.~\cite[\S 9.1]{bcn}. Note that $J(n,D)$ is isomorphic to
$J(n,n-D)$; in the following we therefore restrict to $n \geq 2D$. The Johnson graph $J(n,D)$ is characterized as
distance-regular graph by its intersection array unless $n=8$ and $D=2$, in which case there are also three so-called
Chang graphs.

The {\em Grassmann graph} $J_q(n,D)$ has as vertices the $D$-dimensional subspaces of a vector space of dimension $n$
over $GF(q)$. Two subspaces are adjacent if and only if they intersect in a $(D-1)$-dimensional subspace; cf.~\cite[\S
9.3]{bcn}. Note that $J_q(n,D)$ is isomorphic to $J_q(n,n-D)$; again we therefore restrict to $n \geq 2D$. Metsch
\cite{Me95} showed that the Grassmann graphs are determined by the intersection array if $D \neq 2, \frac{n}{2}$, or
$\frac{n-1}{2}$ (for all $q$) and $(D,q) \neq (\frac{n-2}{2},2),
(\frac{n-2}{2},3)$, or $(\frac{n-3}{2},2)$; see also Section
\ref{sec:Metsch}. For $D=2$, the Grassmann graphs are in general
not determined by the intersection array, as the line graph of a
$2$-$((q^n-1)/(q-1),q+1,1)$ design has the same array. Van Dam and
Koolen \cite{DK05} constructed the {\em twisted Grassmann graphs};
these are distance-regular graphs with the same array as the
Grassmann graphs for $n=2D+1, D\geq 2$, see Section
\ref{twistedsection}.

The {\em Hamming graph} $H(D,e)$ is defined on vertex set $X^D$ of
words of length $D$ from an alphabet $X$ of size $e$. Two words are
adjacent if and only if they differ in precisely one position;
cf.~\cite[\S 9.2]{bcn}. The Hamming graph $H(D,e)$ is characterized
by its intersection array unless $e=4$ and $D>1$, in which case
there are also so-called Doob graphs. A {\em Doob graph} is a
cartesian product of cliques of size 4 and Shrikhande graphs. The
Hamming graph $H(D,2)$ is also called a {\em (hyper)cube} or the
{\em $D$-cube}. Its halved graph is called a {\em halved cube}
$\frac12 H(D,2)$ and is characterized by its intersection array
(see \cite[\S 9.2.D]{bcn}).

The {\em bilinear forms graph} $Bil(D \times e,q)$ has as vertices all $D \times e$ matrices with entries from the
field $GF(q)$, where two matrices are adjacent if and only if their difference has rank $1$; cf.~\cite[\S 9.5.A]{bcn}
or \cite{D}. We shall assume $D \leq e$ in the following, so that $D$ is the diameter. The bilinear forms graph can be
considered as the $q$-analogue of the Hamming graph (view the vertices of the latter as the maps from a set of size $D$
to a set of size $e$), hence also the notation $H_q(D,e)$ is used in the literature. The bilinear forms graph has an
alternative description on the $D$-dimensional subspaces of a $(D+e)$-dimensional vector space that intersect a fixed
$e$-dimensional subspace trivially, where two such subspaces are adjacent if they intersect in a $(D-1)$-dimensional
subspace; this shows that it is isomorphic to a subgraph of the Grassmann graph $J_q(D+e,D)$. Rifa and Zinoviev
\cite{RiZi11} showed that the bilinear forms graph is also a quotient (as defined in Section \ref{sec:CRC}) of the
Hamming graph. Metsch \cite{Me99} showed that the bilinear forms
graph $Bil(D \times e,q)$ is characterized by its intersection
array if $q=2$ and $e \geq D + 4$ or $q \geq 3$ and $e \geq D + 3$;
see also Section \ref{sec:Metsch}. Gavrilyuk and Koolen
\cite{GKbilinear15} extended this characterization with the case
$q=2$ and $e=D$.


The {\em alternating forms graph} $Alt(n,q)$ has as vertices all $n \times n$ skew-symmetric matrices with zero
diagonal and entries from $GF(q)$. Two matrices are adjacent if and only if their difference has rank $2$. Note that a
skew-symmetric matrix has even rank; cf.~\cite[\S 9.5.B]{bcn} or \cite{DG}.

The {\em Hermitian forms graph} $Her(D,q^2)$ has as vertices the $D \times D$ Hermitian matrices with entries in
$GF(q^2)$, i.e., matrices $H$ such that $H_{ij}=(H_{ji})^q$ for all $i$ and $j$. Two matrices are adjacent if and only
if their difference has rank $1$; cf.~\cite[\S 9.5.C]{bcn}. The Hermitian forms graphs are determined by their
intersection arrays for $D \geq 3$, see Section \ref{sec:recentclassical}.

The {\em quadratic forms graph} $Qua(n,q)$ has as vertices the quadratic forms in $n$ variables over $GF(q)$. In the
quadratic forms graph two forms are adjacent if and only if the rank of their difference equals $1$ or $2$;
cf.~\cite[\S 9.6]{bcn} or \cite{E}. Under the group of invertible linear transformations of variables, the quadratic
forms fall into $2n+1$ ($q$ odd) or $\lceil \frac{3n+1}2 \rceil$ ($q$ even) orbits: each form of rank $k \neq 0$ is of
one of two types. For even rank there is the well-known distinction between hyperbolic and elliptic forms; in the case
of odd rank, a (parabolic) form is equivalent to $x_1x_2+\cdots+x_{k-2}x_{k-1}+cx_k^2$, for some $c$, and the type
depends on whether $c$ is a square or not (cf.~\cite[Ch.~IV]{N}). If $q$ is even then each field element is a square,
hence there is no distinction for odd rank.

The {\em dual polar graphs}\footnote{These graphs already appear as
distance-transitive graphs in disguise in a paper by Hua
\cite{Hua1945} from 1945.} have as vertices the maximal isotropic
($D$-dimensional) subspaces of one of the below vector spaces $V$
endowed with a non-degenerate quadratic form. Two subspaces are
adjacent if and only if they intersect in a $(D-1)$-dimensional
space; cf.~\cite[\S 9.4]{bcn}. The following dual polar graphs can
be distinguished:

\bigskip

$\C_D(q)$ for $V=GF(q)^{2D}$ with a symplectic form; $e=1$;

$\B_D(q)$ for $V=GF(q)^{2D+1}$ with a quadratic form; $e=1$;

$\D_D(q)$ for $V=GF(q)^{2D}$ with a quadratic form of Witt
index $D$; $e=0$;

$^2\D_{D+1}(q)$ for $V=GF(q)^{2D+2}$ with a quadratic form of
Witt index $D$; $e=2$;

$^2\A_{2D}(\sqrt{q})$ for $V=GF(q)^{2D+1}$ with a Hermitian
form; $e=\frac{3}{2}$;

$^2\A_{2D-1}(\sqrt{q})$ for $V=GF(q)^{2D}$ with a Hermitian
form;
$e=\frac{1}{2}$.

\bigskip

\noindent Here the mentioned parameter $e$ is related to the classical
parameter $\beta$ of the next section (see Table \ref{tableclassical}).

The dual polar graphs $^2\A_{2D-1}(\sqrt{q})$ are determined by
their intersection arrays for $D \ge 4$, see Section
\ref{sec:recentclassical}. The dual polar graphs $\B_D(q)$ and
$\C_D(q)$ have the same intersection array but are non-isomorphic
unless $q$ is even. The dual polar graph $\D_D(q)$ is the extended
bipartite double of $\B_{D-1}(q)$, and its halved graph, called a
{\em half dual polar graph} $\D_{D,D}(q)$, is the distance
$1$-or-$2$ graph of $\B_{D-1}(q)$. The extended bipartite double of
$\C_{D-1}(q)$ is also distance-regular and is called a {\em
Hemmeter graph}
\cite{BH1992EJC}; its halved graph is the distance $1$-or-$2$ graph
of $\C_{D-1}(q)$ and is called an {\em Ustimenko graph}
\cite{IMU1989EJC}.


\subsubsection{Classical parameters}\label{sec:classicalparameters}

The `classical' distance-regular graphs from the previous section
have intersection numbers that can be expressed in terms of four
parameters, that is, diameter $D$ and numbers $b$, $\alpha$,
$\beta$, in the following way:
%
\begin{align} \label{bb}
b_i & = \bigg(\gauss{D}{1} - \gauss{i}{1}\bigg)
          \bigg(\beta - \alpha \gauss{i}{1}\bigg)
             ~~(i =0,1,\dots, D-1),\notag \\[-5mm]  \\
c_i & = \ \gauss{i}{1}\bigg(1 + \alpha \gauss{i - 1}{1}\bigg)
            ~~(i =1,2,\dots, D),  \notag
\end{align}
%
where $ \gauss{j}{1} = 1+b+b^2 +\cdots +b^{j-1} $ is a Gaussian binomial
coefficient. Therefore, a distance-regular graph is said to have classical
parameters $(D,b,\alpha,\beta)$ if its intersection numbers can be expressed as
in (\ref{bb}). We note that the parameter $b$ must be an integer not equal to 0
or $-1$. The classical examples of distance-regular graphs from the previous
section have classical parameters as in Table \ref{tableclassical} (note that
one family of dual polar graphs has intersection numbers that can be expressed
in two ways). More basic information on distance-regular graphs with classical
parameters can be found in \cite[Ch.~6,~9]{bcn}. Important to mention is that
distance-regular graphs with classical parameters must be $Q$-polynomial. In
Section \ref{sec:Qpol}, we will therefore include also some results on
distance-regular graphs with classical parameters.


\begin{table}%[here]
\begin{center}
    \begin{tabular}{|l|c|c|c|c|}
    \hline
    \ \ \ \ \    & $D$ & $b$ & $\alpha$& $\beta$ \\
    \hline
    Johnson graph $J(n,D), n \geq 2D$ & $D$  & $1$ & $1$     & $n-D$   \\
    \hline
    Grassmann graph $J_q(n,D)$, $n \geq 2D$;  & $D$  & $q$ & $q$
    &$\frac{q^{n-D+1}-1}{q-1}-1$\\        %$\gauss{n-D+1}{1}-1$\\
    twisted Grassmann graph ($n=2D+1$)&&&&\\
    \hline
    Hamming graph $H(D,e)$; & $D$              & $1$ & $0$     & $e-1$   \\
     Doob graph ($e=4$)&&&&\\
    \hline
    Halved Cube $\frac12 H(n,2)$&$\lfloor \frac n2 \rfloor$ &$1$&$2$&$2\lceil \frac n2 \rceil -1$ \\
        \hline
    Bilinear forms graph $Bil(D \times e,q)$,   & $D$              & $q$ & $q - 1$ & $q^e-1$ \\
    $D\leq e$&&&& \\
    \hline

    Alternating forms graph $Alt(n,q)$,&$\lfloor \frac n2\rfloor$
                                              &$q^2$& $q^2-1$ & $q^m-1$ \\
    $m=2\lceil \frac n2 \rceil -1$&&&& \\
    \hline
    Hermitian forms graph $Her(D,q^2)$ & $D$              & $-q$& $-q-1$  & $-(-q)^D-1$
                                                                        \\
    \hline
    Quadratic forms graph $Qua(n,q)$, & $\lfloor \frac{n+1}2\rfloor$
                                             & $q^2$ &$q^2-1$ & $q^m-1$ \\
    $m=2\lfloor \frac n2 \rfloor +1$&&&& \\
    \hline
    Dual polar graph;       &         $D$      & $q$ & $0$     & $q^e$   \\
    Hemmeter graph ($e=0$);&&&&\\
    $^2\A_{2D-1}(\sqrt{q})$ also: & $D$ & $-\sqrt{q}$ &  $\sqrt{q} \frac{1+\sqrt{q}}{1-\sqrt{q}}$  &  $\sqrt{q} \frac{1+(-\sqrt{q})^D}{1-\sqrt{q}}$  \\
    \hline
    Half dual polar graph $\D_{n,n}(q)$,&$\lfloor \frac n2 \rfloor$&$q^2$&$q^2+q$&$\frac{q^{m+1}-1}{q-1}-1$\\
    $m=2\lceil \frac n2 \rceil -1$;&&&&\\
    Ustimenko graph&&&&\\
    \hline
    \end{tabular}\vspace{3mm}
     \caption{Classical parameters of families of distance-regular graphs with unbounded diameter}\label{tableclassical}
\end{center}
\end{table}

\subsubsection{Other families with unbounded diameter}\label{sec:unboundeddiameter}
One of the ultimate problems in this area is to classify the
families of distance-regular graphs with unbounded diameter.
Besides the above known families of distance-regular graphs with
classical parameters and the {\em polygons} (see Section
\ref{sec:polygons}), also the below six families are known. All of them are related to the classical
ones, but they do not have classical parameters themselves.

The {\em folded cube} is obtained by folding the hypercube
$H(n,2)$. Unless $n=6$, it is determined by its intersection array.
For $n=6$, every graph with the relevant intersection array is the
incidence graph of a symmetric $2$-$(16,6,2)$ design. This gives
two other distance-regular graphs (see \cite[\S 9.2.D]{bcn}).

For $n$ even, the folded cube is still bipartite (and the halved
cube is still antipodal). Its halved graph is the {\em folded
halved cube} and it is determined by its intersection array for $n
\geq 12$ (that is, when its diameter is at least $3$; see Section
\ref{sec: partition graphs}).

The Johnson graph $J(2n,n)$ is antipodal, and its folding is called
a {\em folded Johnson graph}. This folded graph is determined by
its intersection array for $n \geq 6$ (that is, when its diameter
is at least $3$; see Section
\ref{sec: partition graphs}).

The folded cube, folded halved cube, and folded Johnson graph are
so-called {\em partition graphs} and these are known to be
$Q$-polynomial (see \cite[\S 6.3]{bcn}).

In Section \ref{sec:oddgraphs}, we already described the {\em Odd
graphs}, which are determined by their intersection array by
Proposition \ref{prop:odd}. The Odd graph is the distance-$D$ graph
of the Johnson graph $J(2D+1,D)$, and it is $Q$-polynomial.

Also the bipartite double of the Odd graph, the {\em Doubled Odd
graph}, is determined by its intersection array (see \cite[\S
9.1.D]{bcn}), but it is not $Q$-polynomial.

The final known family of distance-regular graphs with unbounded
diameter is the family of {\em Doubled Grassmann graphs}. This
graph is the bipartite double of the distance-$D$ graph of the
Grassmann graph $J_q(2D+1,D)$. Like the Doubled Odd graph, it is
determined by its intersection array (see Section
\ref{sec:otherinfinite}), and it is not $Q$-polynomial.

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

\subsection{New constructions}\label{sec:newconstructions}

In this section, we mention some relatively new constructions
of distance-regular graphs.

\subsubsection{The twisted Grassmann graphs} \label{twistedsection}

Van Dam and Koolen \cite{DK05} constructed the first family of
non-vertex-transitive distance-regular graph with unbounded diameter. These
graphs have the same intersection array as certain Grassmann graphs, and are
constructed as follows. Let $q$ be a prime power, and let $D \geq 2$ be an
integer. Let $W$ be a $(2D+1)$-dimensional vector space over $GF(q)$, and let
$H$ be a hyperplane in $W$. Vertices are the $(D+1)$-dimensional subspaces of
$W$ that are not contained in $H$, and the $(D-1)$-dimensional subspaces of
$H$. Two vertices of the first kind are adjacent if they intersect in a
$D$-dimensional subspace; a vertex of the first kind is adjacent to a vertex of
the second kind if the first contains the second; and two vertices of the
second kind are adjacent if they intersect in a $(D-2)$-dimensional subspace.
This graph is distance-regular with the same intersection array as the
Grassmann graph $J_q(2D+1,D)$. In fact, this Grassmann graph and the twisted
Grassmann graph are the point graph and line graph, respectively, of a partial
linear space whose points are the $D$-dimensional subspaces of $W$, and where
a $(D+1)$-dimensional subspace of $W$ that is not contained in $H$ is incident
to the $D$-dimensional subspaces that it contains, and a $(D-1)$-dimensional
subspace of $H$ is incident to the $D$-dimensional subspaces of $H$ containing
it.

The twisted Grassmann graph is not vertex-transitive (it has two orbits of vertices), and hence it is not isomorphic to
the Grassmann graph. Fujisaki, Koolen, and Tagami \cite{FKT07} showed that the automorphism group of the twisted
Grassmann graphs is $P \G L(2D+1,q)_{2D}$, the subgroup of $P \G L(2D+1,q)$ that fixes $H$. Bang, Fujisaki, and
Koolen \cite{BFK09} determined the spectra of the local graphs, and studied in some detail its Terwilliger algebras (as
defined in Section \ref{sec:Talgebra}). Remarkably, these algebras with respect to vertices in distinct orbits are not
the same. The twisted Grassmann graphs are also counterexamples to two conjectures by Terwilliger
\cite[p.~207-210]{Talgebra92}, see \cite{BFK09}. Jungnickel and Tonchev \cite{JuTo09} constructed designs that are
counterexamples for Hamada's conjecture. Munemasa and Tonchev \cite{MuTo09} showed that the twisted Grassmann graphs
are isomorphic to the block graphs of these designs.
Munemasa \cite{Munemasa2014SHCC} showed that the twisted Grassmann graphs can also be obtained from the Grassmann graphs by Godsil-McKay switching (cf.~\cite[\S1.8.3]{BrHa}).


\subsubsection{Brouwer-Pasechnik and Kasami graphs}\label{sec:Kasami}

For prime powers $q$, Pasechnik \cite{BP2011} constructed a distance-regular
graph with intersection array
$\{q^3,q^{3}-1,q^{3}-q,q^3-q^2+1;1,q,q^{2}-1,q^3\}$ as a subgraph of the dual
polar graph $\D_4(q)$; in particular, the induced subgraph on the set of
vertices at maximal distance from an edge.

Brouwer \cite{BP2011} constructed related distance-regular graphs with
intersection array $\{q^{3}-1,q^{3}-q,q^3-q^2+1;1,q,q^{2}-1\}$ as follows.
Consider the vector space $GF(q)^3$ equipped with a cross product $\times$. The
vertex set is $(GF(q)^3)^2$, where a pair $(u,v)$ is adjacent to a distinct
pair $(u',v')$ if and only if $u'=u+v\times v'$. The extended bipartite doubles
of these graphs are the above mentioned graphs constructed by Pasechnik. In
fact, Brouwer's graph is a subgraph of the dual polar graph $\B_3(q)$; in
particular, the induced subgraph on the set of vertices at maximal distance
from a vertex, see \cite{BP2011}.

For even $q$, the mentioned graphs have the same intersection arrays as certain Kasami graphs, cf.~\cite[Thm.~11.2.1
(11),(13)]{bcn}. Pasini and Yoshiara \cite{PY01} constructed distance-regular graphs with the same intersection array
as (bipartite, diameter 4) Kasami graphs using dimensional dual hyperovals. Also the symmetric bilinear forms graphs
for $q$ even and $n=3$ are distance-regular with the same intersection array as (diameter 3) Kasami graphs,
cf.~\cite[p.~285-286]{bcn} and \cite{BCNcoradd}.

Van Dam and Fon-Der-Flaass used almost bent functions to generalize the Kasami graphs, cf.~\cite{EF00},
\cite[Con.~3]{EF03}: Let $W$ be an $n$-dimensional vector space over $GF(2)$, and $f$ be an almost bent function on $W$
with $f(0) = 0$. Then the graph with vertex set $W^2$, where two distinct vertices $(x, a)$ and $(y, b)$ are adjacent
if $a + b = f (x + y)$ is distance-regular with intersection array $\{2^n - 1, 2^n - 2, 2^{n-1} + 1; 1, 2, 2^{n-1} -
1\}$. Recently, a lot of new almost bent functions have been discovered in the guise of quadratic almost perfect
nonlinear functions in odd dimensional vector spaces over $GF(2)$, cf.~\cite{BBMM08, BCL09, Edel}.




\subsubsection{De Caen, Mathon, and Moorhouse's Preparata graphs and crooked
graphs}\label{sec:Preparata}

De Caen, Mathon, and Moorhouse \cite{CMM95} constructed distance-regular
antipodal $2^{2t-1}$-covers of the complete graph $K_{2^{2t}}$, i.e., with
intersection array $\{2^{2t}-1,2^{2t}-2,1;1,2,2^{2t}-1\}$. These graphs are
defined as follows. Consider the vertex set $V=GF(2^{2t-1})\times GF(2) \times
GF(2^{2t-1})$, and let two vertices $(x,i,a)$ and $(y,j,b)$ be adjacent if
$$a+b=x^2y +xy^2 +(i+j)(x^3+y^3).$$
The construction is a bit more general, cf.~\cite{CMM95}, and is related to the Preparata codes. The construction also
allows for taking quotients. In this way, distance-regular graphs with intersection arrays
$\{2^{2t}-1,2^{2t}-2^h,1;1,2^h,2^{2t}-1\}$ for $h=1,2,\dots,2t$ arise. Prior to this construction, no distance-regular
graphs with these intersection arrays were known for $h<t$.

It is noteworthy that the Kasami graphs of the previous section are
induced subgraphs of the Preparata graphs. Because of this
relation, it is not surprising that variations of the above
construction are possible. To obtain these, De Caen and
Fon-Der-Flaass \cite{DeCaenFonDerFlaass} used Latin squares,
whereas Bending and Fon-Der-Flaass \cite{BF98} and Van Dam and
Fon-Der-Flaass \cite{EF03} used highly nonlinear functions such as
crooked functions and almost bent functions with accomplices: Let
$W$ be an $n$-dimensional vector space over $GF(2)$, and $f$ be a
crooked function on $W$. Then the (crooked) graph with vertex set
$W \times GF(2) \times W$, where two distinct vertices $(x,i, a)$
and $(y,j, b)$ are adjacent if $a + b = f (x + y)
+(i+j+1)(f(x)+f(y))$ is distance-regular with the same intersection
array as the Preparata graphs. Godsil and Roy
\cite{GodsilRoy08} determined that the above equation defines a
distance-regular graph precisely when $f$ is crooked. The Gold
functions, given by $f(x)=x^{2^e+1}$ on $GF(2^n)$ with
$\gcd(e,n)=1$ and $n=2t-1$, give the Preparata graphs.

It follows from the observations in \cite[p.~92]{EF03} that bijective quadratic almost perfect nonlinear functions
(that map 0 to 0) are crooked. A new family of such functions was thus constructed by Budaghyan, Carlet, and Leander
\cite[Prop.~1]{BCL08}. See also \cite{Bierbrauer}, but beware that Bierbrauer used a less strict definition of
crookedness (compared to the original one) there.

The paper by De Caen and Fon-Der-Flaass
\cite{DeCaenFonDerFlaass} initiated the prolific construction
by Fon-Der-Flaass \cite{FonDerFlaassprolific} of
distance-regular $n$-covers of complete graphs $K_{n^2}$ by
using affine planes of order $n$. Fon-Der-Flaass realized that
in general, his method produces many (potentially)
non-isomorphic such graphs; at least $2^{\frac12n^3 \log
n(1+o(1))}$ to be more precise. Computational results by
Degraer and Coolsaet \cite{DegraerCoolsaet} confirm this; they
verified that at least 80 of the 94 distance-regular antipodal
$4$-covers of $K_{16}$ can be constructed by Fon-Der-Flaass'
prolific construction. Also the (three) distance-regular
antipodal $4$-covers of $K_{10}$ \cite{DegraerCoolsaet}, the
(two) distance-regular antipodal 3-covers of $K_{14}$
\cite{Deg07}, and the (four) distance-regular antipodal
3-covers of $K_{17}$ \cite{Deg07} were classified by computer
by Degraer and Coolsaet. We also remark that Muzychuk
\cite{Muzy07} extended Fon-Der-Flaass' ideas further.

Godsil and Hensel \cite{GodsilHensel92} (see also \cite{CMM95}) described a
relation between regular antipodal covers of complete graphs and generalized
Hadamard matrices. By constructing skew generalized Hadamard matrices, Klin and
Pech \cite{KlinPech} thus constructed new infinite families of distance-regular
antipodal covers of complete graphs. Their paper contains a good overview of
the state of the art concerning such covers, and has many interesting ideas and
connections. For more background on antipodal covers of complete graphs, we
also refer to Godsil and Hensel \cite{GodsilHensel92} and Godsil
\cite{Godsil96}. For the classification of distance-transitive antipodal covers
of complete graphs, we refer to the paper by Godsil, Liebler, and Praeger
\cite{GLP98}.




\subsubsection{Soicher graphs and Meixner graphs}
\label{sec: Soicher and Meixner}

Soicher \cite{So93} obtained three distance-regular graphs of
diameter four, each being a triple cover of a strongly regular
graph. The first has intersection array $\{416,315,64,1;\linebreak
1,32,315,416\}$, and is a triple cover of the Suzuki graph. The
second has intersection array $\{56,45,16,1;1,8,45,56\}$, and is a
triple cover of the second subconstituent of the McLaughlin graph.
In an unpublished manuscript, Brouwer \cite{AEBSoicher} (see also
\cite{BCNcoradd}) showed that this cover is the only cover of
the second subconstituent of the McLaughlin graph, hence it is the
only graph with the given intersection array. The third cover
constructed by Soicher is the second subconstituent of the second
one, it has intersection array $\{32,27,8,1;1,4,27,32\}$, and is a
triple cover of the Goethals-Seidel graph (the second
subconstituent of the second subconstituent of the McLaughlin
graph). Soicher \cite{Soicher15} also showed that this graph is the
only graph with the given intersection array.

Meixner \cite{Meixner91} implicitly constructed two distance-transitive antipodal covers with intersection arrays
$\{176,135,36,1;1,12,135,176\}$ and $\{176,135,24,1; 1,24,135,176\}$, as the collinearity graphs of the
geometries in \cite[Prop.~4.3]{Meixner91}, see also \cite{BCNcoradd}. Juri\v{s}i\'{c} and Koolen \cite{JuKopre} showed
that the antipodal Meixner $4$-cover is uniquely determined by its intersection array.

Munemasa observed that the Meixner $2$-cover is the extended $Q$-bipartite double of the Moscow-Soicher graph of the
next section, cf.~\cite[Ex.~3.4]{MMW07}.



\subsubsection{The Koolen-Riebeek graph and the Moscow-Soicher graph}\label{sec: KoolenRiebeek}

Brouwer, Koolen, and Riebeek \cite{BKR98} gave a construction of a bipartite
distance-regular graph with intersection array $\{45,44,36,5;1,9,40,45\}$ from
the ternary Golay code. Each of its halved graphs is the complement of the
Berlekamp-van Lint-Seidel graph.

Soicher \cite{So95} constructed another distance-regular graph related to one of the Golay codes, in this case the
binary. It has intersection array $\{110,81,12;1,18,90\}$. Farad\v{z}ev, Ivanov, Klin, and Muzychuk
\cite[p.~119]{FKM94} already mentioned the underlying association scheme of this graph without realizing it was metric.
