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

\subsection{Metsch's characterizations}\label{sec:Metsch}

As mentioned before, Metsch characterized most of the Grassmann graphs and
bilinear forms graphs by their intersection arrays. From these intersection
arrays, he recovers the geometric properties of these graphs. An important
ingredient for this is the following proposition, which is used to construct
lines --- large cliques --- that partition the edge set.

\begin{prop}\label{Metschresult-2.2}{\em \cite[Result~2.2]{Me95}}
Let $\mu \geq 1$, $\lambda_1,~\lambda_2$, and $m$ be integers. Assume that
$\G$ is a connected graph with the following properties:
\begin{enumerate}[{\em (i)}]
\item Every two adjacent vertices have at least $\lambda_1$ and at most
    $\lambda_2$ common neighbors,
\item Every two nonadjacent vertices have at most $\mu$ common neighbors,
\item $2\lambda_1-\lambda_2>(2m-1)(\mu-1)-1$,
\item Every vertex has fewer than $(m+1)(\lambda_1+1)-\frac{1}{2}m(m+1)(\mu-1)$ neighbors.
\end{enumerate}
Define a line to be a maximal clique $C$ satisfying $|C|\geq
\lambda_1+2-(m-1)(\mu-1)$. Then every vertex is on at most $m$ lines, and every
two adjacent vertices lie in a unique line.
\end{prop}

\noindent In \cite{Me95}, Metsch used Proposition \ref{Metschresult-2.2} and a characterization of projective incidence
structures by Ray-Chaudhuri and Sprague \cite{RCSprague} (see also \cite[Thm.~9.3.9]{bcn}, and \cite{Cuypers92} for a
generalization by Cuypers) to characterize the Grassmann graphs. The interesting thing is that he hardly required any
of the regularity conditions that follow from the intersection array. The only conditions that Metsch used were the
intersection number $c_2$ and upper and lower bounds on the number $b_0(x)$ of neighbors of a vertex $x$, and on the
number of common neighbors $a_1(x,y)$ of two adjacent vertices $x$ and $y$, and a lower bound on the number $b_2(x,y)$
of vertices $x$ and $y$ at distance two. In weaker form, the characterization is as follows.

\begin{prop} {\em \cite[Thm.~1.1]{Me95}}
Let $q \ge 2$ be an integer, and let $D$ and $n$ be integers satisfying $2D \le
n$. Let $s+1=(q^{n-D+1}-1)/(q-1)$ and $m=(q^{D}-1)/(q-1)$. Let $\G$ be a
connected $ms$-regular graph with the property that every two adjacent vertices
have $a_1=s-1+(m-1)q$ common neighbors and every two vertices at distance two
have $c_2=(q+1)^2$ common neighbors, and such that every two vertices $x$ and
$y$ at distance two have $b_2(x,y)>(m-q-1)(s-q^2-q)$. 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 $(\frac{n-3}{2},2)$, then $q$ is a
prime power and $\G$ is the Grassmann graph $J_q(n,D)$.
\end{prop}

\noindent Building on work by Huang \cite{Huang87} and a characterization of
attenuated spaces by Sprague \cite{Sprague}, Metsch \cite{Me99} also used
Proposition \ref{Metschresult-2.2} to characterize the bilinear forms graphs.

\begin{prop} Let $\G$ be a distance-regular graph with
classical parameters $(D,q,\alpha,\beta)$, where $\alpha=q-1$ and $D \ge 3$.
Suppose that either $q=2$ and $\beta \ge q^{D+4}-1$ or $q \ge 3$ and $\beta \ge
q^{D+3}-1$. Then $q$ is a prime power, $\beta=q^e-1$ for some integer $e$, and
$\G$ is the bilinear forms graph $Bil(D \times e,q)$.
\end{prop}

\noindent Proposition \ref{Metschresult-2.2} can be used further in
characterizing other geometric distance-regular graphs. We will get back to
this in Section \ref{sec:geometric}.


\subsection{Characterization of Doubled Odd and Doubled Grassmann
graphs}\label{sec:otherinfinite} In Section \ref{sec:generalizations+geometric}
we mentioned the distance-biregular graphs that arise as incidence graphs
between the vertices and the cliques coming from the $(D+1)$-subspaces in the
Grassmann graph $J_q(n,D)$, and the similar one (with $(D+1)$-subsets) from the
Johnson graph $J(n,D)$. Cuypers \cite{Cuypers92} classified the
distance-biregular graphs with diameter at least 5 and $c_2^R=1<c_3^R=c_4^R$
($R$ being one of the color classes): the only ones are the above mentioned
graphs and the Doubled Moore graphs. This implies that the Doubled Grassmann
graphs, the Doubled Odd graphs (the case $n=2D+1$), and also the Doubled
Hoffman-Singleton graph are determined as distance-regular graphs by their
intersection arrays.

Hiraki \cite{Hi07} (also) characterized the Odd graphs and the Doubled Odd
graphs among the distance-regular graphs by a few of their intersection
numbers, as we already mentioned for the Odd graphs in Section
\ref{sec:determinedbyarray}. Moreover, Hiraki \cite{Hi03D} characterized the
Doubled Grassmann graphs, the Doubled Odd graphs, and the Odd graphs by their
strongly closed subgraphs.




\subsection{Bounds on claws}\label{sec:claws}
An $m$-claw in a graph $\G$ is an induced $m$-star $K_{1,m}$, or in other
words, a coclique of size $m$ in one of the local graphs $\Upsilon(x), x \in
V$.

If $\G$ is a geometric distance-regular graph (with respect to a set of
Delsarte cliques $\cal{C}$) with smallest eigenvalue $-m$, then it follows
easily that each vertex is in $m$ cliques of $\cal{C}$, and hence $\G$ has
no $(m+1)$-claws. Under some conditions, a reverse statement can be made, as we
shall see at the end of this section. Besides this, the existence of claws of
certain size gives rise to new parameter conditions. But as we shall see,
sometimes the intersection numbers force the existence of claws, thus giving
some nonexistence results.

A special case of a result of Metsch's work \cite[Lemma~1.1.b]{metsch91} on the existence of large cliques in graphs is
the following (see also work by Godsil \cite[Lemma~2.3]{Godsil93} or Koolen and Park \cite[Lemma~2]{KoPa10}).

\begin{lemma}
Let $\G$ be a distance-regular graph. Let $x$ be a vertex and let $\mu$ be
the maximum size of $\G(x) \cap \G(y) \cap \G(z)$ where $y \sim x
\sim z$ and $y \not\sim z$. If the local graph $\Upsilon(x)$ contains a
coclique of size $m$, then $\mu \geq \frac{ m (a_1 +1) - k}{\binom{m}{2}}$.
\end{lemma}

\noindent The following consequence of this lemma was observed by Koolen and Park \cite[Thm.~4]{KoPa10}, using that a
`greedy' coclique in $\Upsilon(x)$ has at least $k/(a_1+1)$ vertices.

\begin{prop}\label{shillaterw}
Let $\G$ be a distance-regular graph with valency $k$ and diameter $D \geq
2$, and let $m' = \lceil \frac{k}{a_1+1} \rceil.$ Then
\begin{equation}\label{KoPa}
	c_2 -1 \geq \frac{m'(a_1+1)-k}{\binom{m'}{2}}
\end{equation}
with equality implying that $\G$ is a Terwilliger graph.
\end{prop}

\noindent This result shows that there are no distance-regular graphs with
intersection arrays $\{ 44, 30, 5; 1, 3, 40\}$, $\{65, 44, 11; 1, 4, 55\}$, $\{
81, 56, 24, 1; 1, 3, 56, 81\}$, $\{117, 80, 30, 1; 1, 6, 80, \linebreak 117\}$, $\{ 117,
80, 32, 1; 1, 4, 80, 117\}$ and $\{ 189, 128, 45, 1; 1, 9, 128, 189\}$ (the
last four were also ruled out by Juri\v{s}i\'{c} and Koolen \cite{JuKo00EuJC}).

Gavrilyuk \cite{GavElec10} showed that the only distance-regular graphs with $c_2>1$ for which equality holds in
\eqref{KoPa} are the Icosahedron, the Conway-Smith graph, and the Doro graph. Gavrilyuk \cite{Gav11} also extended the
above by using Brooks' theorem to eliminate the existence of a distance-regular graph with intersection array $\{ 55,
36, 11; 1, 4, 45\}$. Brooks' theorem (see \cite[Thm.~14.4]{BondyMurty}) states that the chromatic number of a connected
graph is at most its maximum valency, except for the odd cycles and complete graphs. For a distance-regular graph
$\G$, this implies that $\Upsilon(x)$ has a coclique of size at least $k/a_{1}$, unless possibly when $\Upsilon(x)$
contains an odd cycle (if $a_1=2$) or an $(a_1+1)$-clique as one of its components. This means that Proposition
\ref{shillaterw} can be sharpened a bit.

\begin{prop} Let $\G$ be a distance-regular graph with valency $k$ and
diameter $D \geq 2$, and let $m'' =\lceil \frac{k}{a_1} \rceil$. If for some
vertex $x$, the local graph $\Upsilon(x)$ does not contain an odd cycle or an
$(a_1+1)$-clique as one of its components, then $$c_2 -1 \geq
\frac{m''(a_1+1)-k}{\binom{m''}{2}}.$$
\end{prop}

\noindent In particular, if $\G$ has smallest eigenvalue $\theta_{\min}$,
then this proposition can be applied when $-1- \frac{b_1}{\theta_{\min}+1} <
a_1$ (in which case the local graph is connected) and $a_1 \neq 2$.

In her work on distance-regular graphs without $4$-claws, Bang \cite{Bapre} also ruled out the intersection array $\{
55, 36, 11; 1, 4, 45\}$. Moreover, she related such graphs to geometric distance-regular graphs with smallest
eigenvalue $-3$. This led to the following more general result by Bang and Koolen \cite{BaKo??}.

\begin{prop}
Let $m \geq 3$ be an integer, and let $\G$ be a distance-regular graph with
diameter $D \geq 2$ and valency larger than $\max\{m^2-m, \frac{m^2-1}m(a_1
+1)\}.$ Then $\G$ has no $(m+1)$-claws if and only if $\G$ is a
geometric distance-regular graph with smallest eigenvalue $-m$.
\end{prop}

\noindent For $m=3$, a slightly stronger result was obtained by Bang \cite{Bapre}, in the sense that the
corresponding result holds for valency larger than $\max\{3, \frac83(a_1 +1)\}.$ We finally note that the
distance-regular graphs without $3$-claws have been determined by Blokhuis and Brouwer \cite{BlBr97},
and that some more work on distance-regular graphs without $4$-claws has been done by Guo and
Makhnev \cite{GM2013} and Bang, Gavrilyuk, and Koolen \cite{BGK4claws}.



\subsection{Sufficient conditions}\label{sec:geometric}

\begin{prop}\label{suffgeom1}
Let $\G$ be a distance-regular graph with diameter $D$ with the property
that there exists a positive integer $m$ and a set $\cal C$ of cliques in
$\G$ such that every edge is contained in exactly one clique of $\cal C$
and every vertex $x$ is contained in exactly $m$ cliques of $\cal C$. If
$|{\cal C}| < |V|$, then $\G$ is geometric with smallest eigenvalue $-m$.
In particular, this is the case if $\min\{ |C| : C \in {\cal C}\} > m$.
\end{prop}
\begin{proof}
Consider the $|V| \times |{\cal C}|$ incidence matrix $N$, where $N_{xC} =
1$ if $x \in C$ and 0 otherwise. Then $NN^{\top} = m I + A$, so the smallest
eigenvalue  $\theta_{\min}$ of $\G$ satisfies $\theta_{\min} \geq -m$.
Suppose now that $| V| > |{\cal C}|$. Then $NN^{\top}$ is singular and hence
$\theta_{\min} = -m$. By the Delsarte bound, every clique $C$ has size at most
$1 -\frac{k}{\theta_{\min}}= 1 + \frac{k}{m} $. On the other hand, by
considering the cliques $C  \in {\cal C}$ containing a fixed vertex, we see
that they have $1 + \frac{k}{m}$ vertices on average. This means that all
cliques in $C$ contain exactly $1 + \frac{k}{m}$ vertices, and hence $\G$
is a geometric distance-regular graph. In particular, if $\min\{ |C| : C \in
{\cal C}\} > m$, then it follows by counting the number of incident pairs
$(x,C)$ in two different ways that $|{\cal C}| < |V|$, so $\G$ is
geometric.
\end{proof}

\noindent This proposition implies that distance-regular graphs
of order $(s,t)$ are geometric with smallest eigenvalue $-t-1$ if
$s>t$; see also Corollary \ref{s>t implies geometric}. Using
Proposition \ref{Metschresult-2.2} we now obtain the following
result.

\begin{prop}\label{prop@} Let $m \geq 2$ be an integer, and let $\G$ be a
distance-regular graph with $(m-1)(a_1+1) < k < m(a_1 +m)$ and
diameter $D \geq 2$. If $a_1 \geq \frac12m(m+1)(c_2+1) $, then
$\G$ is geometric with smallest eigenvalue $-m$.
\end{prop}
\begin{proof}
The two given lower bounds on $a_1$ assure that Proposition
\ref{Metschresult-2.2} can be applied, i.e., that $ a_1 \geq (2m-1)(c_2-1)$ and
$k<(m+1)(a_1+1)-\frac12m(m+1)(c_2-1)$. Thus the set ${\cal C}$ of maximal
cliques of size at least $a_1 +2 - (m-1)(c_2-1)$ forms a set of lines such that
each vertex is in at most $m$ lines, and each edge is in exactly one line. The
given lower bound on $k$ assures that each vertex is in exactly $m$ lines.
Because the minimal line size is at least $a_1 +2 - (m-1)(c_2-1)$, which is at
least $m+1$ by one of the given inequalities, it follows from Proposition
\ref{suffgeom1} that $\G$ is geometric with smallest eigenvalue $-m$.
\end{proof}

\noindent We note that the assumption $k<m(a_1+m)$ holds for all
distance-regular graphs with smallest eigenvalue $-m$ (see \cite{KoBa10}).


\subsection{Distance-regular graphs with a fixed smallest eigenvalue}\label{sec:fixedsmallestev}

Generalizing results by Neumaier \cite{Neu80} and Godsil \cite{Godsil93}, Koolen
and Bang \cite{KoBa10} showed the following.
\begin{theorem}\label{thm:nongeometric}
For given $m \geq 2$, there are only finitely many non-geometric
distance-regular graphs with both valency and diameter at least $3$ and smallest
eigenvalue at least $-m$.
\end{theorem}

\noindent Note that valency $2$ is excluded because of the odd polygons; and diameter $2$ because of the complete
multipartite graphs. Koolen and Bang \cite{KoBa10} did not quite prove this result, as they restricted themselves to
graphs with $c_2 \geq 2$. The graphs with $c_2 =1$ are of order $(s,t)$, with $s=a_1+1$ and $t=k/s-1$. If such a graph
has smallest eigenvalue at least $-m$, then by interlacing (see Section \ref{sec:interlacing}), the existence of a
$(t+1)$-claw implies that $t+1 \leq m^2$. Because a distance-regular graph of order $(s,t)$ that is not geometric has
$s\leq t$, it follows that $k=s(t+1) \leq (m^2-1)m^2$. Therefore the result for the case $c_2=1$ follows from the
Bannai-Ito conjecture. Note that the (general) result was known for $m=2$, see \cite[Thm.~3.12.4,~4.2.16]{bcn}.

Because the smallest eigenvalue of a geometric graph is always integral, this theorem also gives a partial answer to
the question from \cite[p.~130]{bcn} whether every distance-regular graph with valency at least three and diameter at
least three has an integral eigenvalue besides the valency.

One may also wonder whether it is true that for a given integer $m \geq 2$,
there are only finitely many geometric distance-regular graphs with
$D \geq 3$, $c_2 \geq 2$, and smallest eigenvalue $-m$, besides the Grassmann graphs, Johnson graphs, bilinear forms
graphs, and Hamming graphs.
For $D=2$ this is not true, but Neumaier \cite{Neu80} showed that
in essence the geometric strongly regular graphs fall into two infinite classes.





Concerning large smallest eigenvalue, we know that the distance-regular graphs with smallest eigenvalue $-1$ are
exactly the complete graphs. The ones with smallest eigenvalue $-2$ are either strongly regular (and classified by
Seidel \cite{Seidel-2}) or line graphs (and classified by Mohar and Shawe-Taylor \cite{Mdrline})
\cite[Thm.~3.12.14]{bcn}. Among these, the only geometric distance-regular graphs with $D \geq 3$ and $k \geq 3$ are
the generalized $2D$-gons of order $(s, 1)$, $s \geq 2$ and $D=3,4,6$, and the line graphs of the Petersen graph, the
Hoffman-Singleton graph, and putative Moore graphs on 3250 vertices (see \cite[Thm.~4.2.16]{bcn}).
Bang and Koolen \cite{BaKo14} finished the classification of the geometric distance-regular graphs with diameter at least $3$, $c_2 \geq 2$, and smallest eigenvalue $-3$, by showing that such a graph is a Hamming graph, Johnson graph, or a generalized quadrangle of order $(s,3)$ minus a spread (with $s=3$ or $5$).
Yamazaki \cite{Y95} obtained strong restrictions on distance-regular graphs of
order $(s,2)$, $s \geq 3$. The (five) distance-regular graphs of order $(2,2)$ were classified by Hiraki, Nomura, and
Suzuki \cite{HNS}. All of these graphs are geometric with smallest eigenvalue $-3$.


\subsection{Regular near polygons}\label{sec:rnp}

Recall from Section \ref{sec:generalizations+geometric} that a distance-regular
graph $\G$ of order $(s,t)$ with diameter $D$ is called a regular near
$2D$-gon if $a_i = c_i a_1$ for all $i=1,2, \ldots, D$, and that such a graph is
geometric. We call $\G$ thick if $s \geq 2$.

\begin{theorem}\label{thm:RNPthick}{\em (cf.~\cite[Thm.~6.6.1,~9.4.4]{bcn})}
Let $\G$ be a thick regular near $2D$-gon with $D \geq 4$. If $c_2 \geq 3$
or $c_i = i \ (i=2,3)$, then $\G$ is either a dual polar graph or a Hamming
graph.
\end{theorem}

\begin{proof}
(sketch) For $c_2 \geq 3$, the proof is implicitly given in \cite{bcn}. Brouwer and Wilbrink \cite{BWilbrink}
showed that a thick regular near $2D$-gon with $D \geq 4$ and $c_2 \geq 3$ satisfies $c_3 = c^2_2 - c_2 +1$ (the gap as
mentioned in \cite[p.~206]{bcn} is repaired by De Bruyn \cite{DBr06a}). From \cite[p.~277,~Rem.~ii]{bcn} (a remark on a
a result by Brouwer and Cohen \cite{bcohen}), it follows that a thick regular near $2D$-gon with $c_3 = c^2_2 - c_2 +1$
and $c_2 \geq 3$ is a dual polar graph.

For the case $c_i = i \ (i=2,3)$, we will give a sketch of the proof, as it is not in the literature. Let $\G$ be a
thick regular near $2D$-gon of order $(s, t)$ with $D \geq 4$ and $c_i = i \ (i=2,3)$. First, by a result of Brouwer
and Wilbrink \cite{BWilbrink}, one may assume that $c_i = i$ for $i \leq D-1$. Second, it can be shown that if $\G$
is of order $(s,t)$ with $c_i = i \ (i=2,3)$ and $a_2=c_2a_1$, then there exists a map $\phi: H(t+1,s-1) \rightarrow
\G$, such that the partition $\{\phi^{-1}(x):x\in V\}$ is completely regular (cf.~\cite[Thm.~3]{Nomura90}). Using
Theorem \ref{uniformlyregularpartition}, one can show that $\phi^{-1}(x)$ is a completely regular code with minimum
distance $2D$. Now its truncated code is a perfect $(D-1)$-error-correcting code and by the perfect code theorem (see
for example \cite{Hong}), the only such codes with $D \geq 4$ (that are relevant to us; $s \geq 2$) are the codes
consisting of exactly one code word. This shows that $c_D = D$ and that $\G$ is the Hamming graph $H(D, s+1)$. This
finishes the proof of the theorem.
\end{proof}

\noindent In some cases, the intersection numbers of a distance-regular graph imply that it must be a regular near
$2D$-gon; if $c_2=1$, $a_1 \leq 1$, or the graph has classical parameters $(D,-a_1-1,\alpha,\beta)$, see
\cite{Terwilliger1995EJC}. This for example implies that there can be no distance-regular graphs with intersection
array $\{147,144,135;1,4,49\}$ (and classical parameters $(3,-3,-3,21)$), because it would yield a regular near hexagon
with $(s,c_2,c_3)=(3,4,49)$ and this was ruled out by Shult according to Brouwer \cite{Brouwer1981}.

Let $\G$ be a thick regular near polygon with diameter $D$ and head $h$. Hiraki \cite{H97} showed that if $D \geq
2h+1$, then $h \in \{1,2, 3\}$ (he mentions also the possibility $h=5$, but this would lead to a thick generalized
12-gon, a contradiction). This result also follows from Proposition \ref{mbounded} (ii) ($m=h-1$), as we obtain a thick
generalized $2(h+1)$-gon as strongly closed subgraph, and by the Feit-Higman theorem (cf.~\cite[Thm.~6.5.1]{bcn}), it
follows that $h+1 \in \{2, 3, 4\}$. Hiraki \cite{Hi499} conjectured that if $D> 2h+1$, then $h =1$. For a thick regular
near $2D$-gon with $D \leq 2h$, one can bound the valency in terms of $a_1$, see \cite{HiKo04c}.

De Bruyn and Vanhove \cite{DeBVhpre} (see also \cite{Vanhove2012JAC}) obtained
that for a regular near $2D$-gon with $a_1>0$, the intersection numbers satisfy
$c_2 \leq (a_1+1)^2+ 1$ and
$$\frac{((a_1+1)^i -1)(c_{i-1} - (a_1 +1)^{i-2})}{(a_1+1)^{i-2} -1} \leq c_{i} \leq
\frac{((a_1+1)^i +1)(c_{i-1}+ (a_1 +1)^{i-2})}{(a_1+1)^{i-2} + 1}$$ for $i=3,4, \ldots, D$. Neumaier \cite{Neu1990JCTA}
obtained the upper bound for odd $i$ and the lower bound for even $i$ as a specialization of the balanced set condition
of Section \ref{sec:Qpolcharacterizations} for the smallest eigenvalue of a regular near $2D$-gon. For $D=i=3$, the
upper bound is the Haemers-Mathon bound \cite[p.~60]{Haemersthesis}\footnote{This bound is also called the Mathon
bound. It was obtained jointly by Haemers and Mathon. Yanushka recognized that the bound can be obtained from a Krein
condition. Besides the remark in \cite{Haemersthesis}, this is all unpublished.} for regular near hexagons. The upper
bound for even $i$ and the lower bound for odd $i$ can be seen as a specialization of Tonejc's \cite{JurTonpre}
modification of the balanced set condition.



The following is a slight extension of a result due to Brouwer, Godsil, Koolen, and Martin \cite[Thm.~10]{BGKM03}:
\begin{prop}\label{thm:BGKM}
Let $\G$ be a thick regular near $2D$-gon with quads (i.e., geodetically closed subgraphs with diameter two).
Then the second smallest eigenvalue $\theta_{D-1}$ of $\G$ satisfies
\begin{equation*}
	\theta_{D-1} \geq a_1+1-\frac{b_1}{(a_1+1)(c_2-1)},
\end{equation*}
with equality if and only if every quad has width and dual degree summing to $D$. Equality occurs only for the dual
polar graphs and Hamming graphs.
\end{prop}


\noindent The last sentence of the above proposition follows from the following. Let $H$ be a subhexagon of $\G$ and
$Q$ be a subquadrangle in $H$. Brouwer and Wilbrink \cite{BWilbrink} showed that $c_3 \geq c_2(c_2-1) +1$ with equality
if and only if there is no vertex at distance 2 from $Q$ in $H$; see also \cite[p.~26]{DBr06a}. Suppose there is a
vertex $x$ at distance 2 from $Q$ in $H$. If $D \geq 4$, this means that $Q$ cannot be a completely regular code in
$\G$, as this vertex has distance at most 3 to all vertices in $Q$, while there also exists a vertex $y$ at distance 2
from $Q$ with distance 4 to some vertex in $Q$. If $D=3$, then the dual degree is at least the covering radius of $Q$
in $H$, which is at least two, and therefore the sum of the width and dual degree is at least 4. Therefore $c_3 =
c_2(c_2-1) +1$, and hence $\G$ is a dual polar graph or a Hamming graph (for $D \geq 4$, this follows from Theorem
\ref{thm:RNPthick}, whereas for $D=3$, it follows from \cite[Thm.~9.4.4]{bcn}).

For more results on regular near polygons, we refer to \cite{HiKo04a, HiKo04b, HiKo06, TW05}.
