% 14-12: a few typos and errors corrected
% 25-12: changed notation in the footnote in Section 11.6

% 14-3: proofread all sections; added a comment on Keevash's result in Section 11.5

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

Let $\G$ be a distance-regular graph.
In this section, a subgraph in $\G$ will always be an induced subgraph.
Recall that a code in $\G$ is simply a non-empty subset of $V_{\G}$.
Therefore, subgraphs, codes, and (vertex) subsets will be virtually the same objects in this section, and we shall adopt one of these names depending on the context.
Completely regular codes will be separately discussed in Section \ref{sec:crc}.


\subsection{Strongly closed subgraphs}\label{sec: strongly closed subgraphs}

Suzuki \cite[Thm.~1.1]{Su295} showed that strongly closed subgraphs of distance-regular graphs are usually
distance-regular.

\begin{theorem}
Let $\Delta$ be a strongly closed subgraph of a distance-regular graph
$\G$. Let $h$ be the head of $\G$ and $k$ be the valency of $\G$. Then one
of the following holds:
\begin{enumerate}[{\em (i)}]
\item $\Delta$ is distance-regular,
\item $2 \leq D_{\Delta} \leq h$,
\item $h$ and $D_{\Delta}$ are even, and $\Delta$ is a distance-biregular graph with $c_{2i-1}
    =c_{2i}$ for all $i=1,2,\dots, \frac{1}{2}D_{\Delta}$,
\item $h =3$, $D_{\Delta}= 5$, and $\Delta$ is isomorphic to the graph obtained by replacing each edge
in a complete graph $K_{\ell+1}$, $\ell \geq 3$, by a path of length $3$,
\item $h =6$, $D_{\Delta} = 8$, and $\Delta$ is isomorphic to the
graph obtained by replacing each edge in a Moore graph with valency $\ell\in \{3,7,57\}$ by a
path of length $3$.
\end{enumerate}
\end{theorem}

\noindent
It follows that (i) holds above precisely when $b_{D_{\Delta}-1}>b_{D_{\Delta}}$, and that (iv) or (v) hold above precisely when $a_1=0$ and $(c_{D_{\Delta}-1},a_{D_{\Delta}-1})=(c_{D_{\Delta}},a_{D_{\Delta}})=(1,1)$.
The Biggs-Smith graph is the only known example of a distance-regular graph with $h\geq 2$ which satisfies $(c_{h+1},a_{h+1})=(c_{h+2},a_{h+2})=(1,1)$.
We note that if $c_2 \geq 2$ then every strongly closed subgraph of $\G$ is distance-regular.

Hiraki \cite{Hi98} introduced the \emph{condition} $\mathrm{(SC)}_m$ as the condition\footnote{We note that $\mathrm{(SC)}_m$ for some $m\in\{1,2,\dots,D_{\G}-1\}$ implies $K_{2,1,1}$-freeness, which in turn implies $h$-boundedness, where $h$ is the head of $\G$; cf.~\cite[p.~129, Remarks]{Hi01}.} that for all vertices $x$ and $y$ at
distance $m$ there exists a strongly closed subgraph $\Delta(x,y)$ with diameter $m$ containing $x$ and $y$.
Hiraki \cite[Thm.~1]{Hi01} showed that $\mathrm{(SC)}_m$ is equivalent to $m$-boundedness for $m=1,2,\dots,D_{\G}-1$.

It is clear that if $\G$ is $m$-bounded then it is $(m+1)$-parallelogram-free.
The converse is not true in general because every bipartite distance-regular graph is parallelogram-free, but it does not even need to be $2$-bounded, as the incidence graph of a $2$-$(11,6,3)$-design shows.
In some cases, however, $(m+1)$-parallelogram-freeness is known to be equivalent to $m$-boundedness.

\begin{prop}\label{mpf}
Let $\G$ be a distance-regular graph with diameter $D$ and let $m\in\{1,2, \ldots,D-1\}$.
Suppose one of the following holds:
\begin{enumerate}[{\em (i)}]
\item $m=1$,
\item $c_2 >1$ and $a_1 >0$,
\item $c_2 =1$ and $a_2 > a_1 >0$,
\item $m=2$ and $a_2 > a_1=0$,
\item $c_{m+1} = 1$ and $a_2 > a_1$.
\end{enumerate}
Then $\G$ is $(m+1)$-parallelogram-free if and only if $\G$ is $m$-bounded.
\end{prop}

\noindent
We remark that (i) is obvious, (ii) was shown by Weng \cite[Thm.~6.4]{W98}, (iii) was shown by Suzuki \cite{S96}, (iv) was
shown by Suzuki \cite{S96} for the case $c_2 =1$ (extending \cite[Lemma 4.3.13]{bcn}) and by Weng \cite[Prop.~6.7]{W98}
for the case $c_2>1$, and (v) was shown by Hiraki \cite{H196}. Hiraki \cite{Hiraki08b} also obtained other sufficient
conditions for a distance-regular graph to be $m$-bounded.

The following proposition summarizes the known results on $\G$ being $m$-bounded for some $m$ that have been obtained by using combinatorial methods.
Some more results are known under the assumption that $\G$ is $Q$-polynomial; cf.~ Section \ref{sec:recentclassical}.
\begin{prop}\label{mbounded}
Let $\G$ be a distance-regular graph with diameter $D \geq 3$ and head
$h\geq 1$. Let $m\in\{1,2,\ldots, D-h\}$. Then $\G$ is $m$-bounded
if one of the following holds:
\begin{enumerate}[{\em (i)}]
\item $c_{m+h} = 1$ and $a_{m-1}<a_m$,
\item $\G$ is $K_{2,1,1}$-free, $a_1>0$, $a_i = c_i a_1$ for $i=1,2, \dots,
    m+h-1$, and $c_{m-1} < c_m$.
\end{enumerate}
\end{prop}

\noindent
Result (i) was obtained by Ivanov and Brouwer (cf.~\cite[Prop.~4.3.11]{bcn}) for $m=2$, and by
Hiraki \cite[Thm.~1.3]{Hi98} for the other cases.
Result (ii) was obtained by Hiraki \cite[Thm.~1.1]{Hi499}, generalizing a result of Brouwer and Wilbrink \cite{BWilbrink} for thick regular near polygons with $h=1$.
We remark that each of the assumptions (i) and (ii) implies $b_{m-1}>b_m$, so that if $x$ and $y$ are at distance $m$ then $\Delta =\Delta(x,y)$ is distance-regular with valency $a_m+c_m$.
In particular, if $c_{2h+1} =1$ and $m = h+1$,
then $\Delta$ is a Moore geometry and it is known that such a graph is either an odd polygon or has diameter at most
$2$; cf.~\cite[Thm.~6.8.1]{bcn}.
This shows the following proposition in the case $a_1 >0$.
The case $a_1 =0$ uses results by Chen, Hiraki, and Koolen \cite{CH99, Hi94,HK02}.

\begin{prop}{\em \cite[Thm.~2]{Hi01}}
Let  $\G$ be a distance-regular graph with head $h \geq 1$ and diameter $D
\geq 2h+3$. Then $h =1$ or $c_{2h+3} \geq 2$.
\end{prop}

\noindent
We note that Wang \cite{W01} did related work. We remark also that if $\G$ is a distance-regular graph with $h=1$ and
$c_4 =1$, then by Proposition \ref{mbounded}(i) and a result from `BCN' \cite[Thm.~5.9.9(i)]{bcn}, $\G$ has a
distance-regular subgraph with diameter 3 and $c_3=1$. No such (latter) graph is known, however. Chen, Hiraki, and
Koolen \cite{CHK98} in fact showed that no such graph with $a_1 \neq 3$ and $a_1 \leq 30$ exists.

Let $\G$ be a distance-regular graph with diameter $D$.
Suppose $\G$ is $D$-bounded and every strongly closed subgraph is regular.
In particular, we have $b_i>b_{i+1}$ for $i=0,1,\dots,D-1$.
Let $\mathscr{S}$ be the poset
consisting of all strongly closed subgraphs of $\G$ with partial order defined by reverse inclusion. Weng
\cite{W97} showed that $\mathscr{S}$ is a ranked meet semilattice and every interval in $\mathscr{S}$ is atomic and
lower semimodular.
He also showed the inequalities
\begin{equation*}
	\frac{b_{D-i-1}-b_{D-i+1}}{b_{D-i-1}-b_{D-i}}\geq \frac{b_{D-i-2}-b_{D-i}}{b_{D-i-2}-b_{D-i-1}} \quad (i=1,2,\dots,D-2),
\end{equation*}
with equality for all $i=1,2,\dots,D-2$ if and only if every interval in $\mathscr{S}$ is a modular atomic lattice.
See also Section \ref{sec: posets}.

For some more work on strongly closed subgraphs in distance-regular graphs, we refer to \cite{Hiraki2012GC} and the references therein.


\subsection{Bipartite closed subgraphs}

Let $\G$ be a distance-regular graph with diameter $D$.
We say the \emph{condition $\mathrm{(BGC)}_j$} holds if for every pair of vertices at
distance $j$ there exists a bipartite closed subgraph with diameter $j$ containing this pair. This condition was introduced by Hiraki \cite{Hi303}, and he showed that $\mathrm{(BGC)}_j$ with $j\in\{1,2,\dots,D-1\}$ implies
$\mathrm{(BGC)}_i$ for all $i=1,2, \ldots, j$.
By combining results of Hiraki \cite{Hi303} and Koolen \cite{Ko192, Ko292}, we have the following.

\begin{prop} {\em \cite[Cor.~4.8]{Hi303}}
Let $\G$ be a distance-regular graph with diameter $D \geq 3$. Let
$t\in\{2,3,\dots, D-1\}$ be such that $c_t = c_{t-1}+1$ and $a_1 = a_2 = \dots =
a_{t-1} = 0$. Then the condition $\mathrm{(BGC)}_t$ holds if and only if one of the
following holds:
\begin{enumerate}[{\em (i)}]
\item $(c_1, c_2, \dots, c_t) = (1, 1, \dots, 1, 2)$ and every bipartite
    closed subgraph with diameter $t$ is the ordinary $2t$-gon,
\item $(c_1, c_2, \dots, c_t) = (1, 2, \dots, t)$ and every bipartite
    closed subgraph with diameter $t$ is the $t$-cube,
\item $t = 2s+1$ is odd, $(c_1, c_2, \dots, c_t) = (1,1, 2,2,\dots, s,s,s+1)$,
    and every bipartite closed subgraph with diameter $t$ is the Doubled Odd
    graph with valency $s+1$,
\item $t=4$, $(c_1, c_2, c_3, c_4) = (1, 1, 2, 3)$, and every bipartite
    closed subgraph with diameter $4$ is the Pappus graph.
\end{enumerate}
\end{prop}

\noindent
Some more general results are obtained by Hiraki \cite{Hi303}.


\subsection{Maximal cliques}

In most cases, it is easy to determine the maximal cliques of classical distance-regular graphs; cf.~\cite{Hemmeter1986EJC}.
However, the structure of the maximal cliques of the quadratic forms graphs turns out to be extremely complicated.
Hemmeter, Woldar, and Brouwer completed the classification of the maximal cliques in this case in a series of papers \cite{Hemmeter1988EJC,HW1990EJCa,HW1990EJCb,BHW1995EJC,HW1999EJC}.
Brouwer and Hemmeter \cite{BH1992EJC} classified the maximal cliques of half dual polar graphs and Ustimenko graphs (which are the distance $1$-or-$2$ graphs of dual polar graphs $\B_m(q)$ and $\C_m(q)$, respectively).
The maximal cliques of twisted Grassmann graphs were described by Van Dam and Koolen \cite{DK05}.

Hemmeter \cite{Hemmeter1986EJC} observed that if $\G$ is a bipartite distance-regular graph with diameter $D\geq 4$, then $\G_1(x)$ is a maximal clique of the halved graph for every $x\in V_{\G}$.
Using this fact, he was able to determine all bipartite distance-regular graphs whose halved graphs belong to one of the known (at the time) infinite families with unbounded diameter; cf.~\cite{Hemmeter1984UM,Hemmeter1988EJC}.
Brouwer, Godsil, Koolen, and Martin \cite[Cor.~2]{BGKM03} showed that if a distance-regular graph $\G$ has a Delsarte clique then it cannot have an antipodal cover of odd diameter.
Van Dam and Koolen \cite{DK05} looked at the structure of the maximal cliques of the twisted Grassmann graphs to show that these graphs are not vertex-transitive.

\subsection{Convex subgraphs}

Lambeck \cite{Lambeck1990D} studied in detail the noncomplete convex subgraphs of classical distance-regular graphs.
He classified such subgraphs in Johnson, Hamming, Grassmann, dual polar, bilinear forms, Hermitian forms, alternating forms graphs, and also quadratic forms graphs $Qua(n,q)$ with $q$ odd.
The noncomplete convex subgraphs of $Qua(n,q)$ with $q$ even were classified by Munemasa, Pasechnik, and Shpectorov \cite{MPS1993JAC}.
It turns out that if $\G$ is one of these graphs then its noncomplete convex subgraphs are distance-regular and belong to the same family as $\G$, with the exception of $Her(D,4)$, which has $K_{2,2}$ as a convex subgraph.

Tanaka \cite{Tanaka2011EJC} used the above results to describe the descendents (cf.~Section \ref{sec: posets}) of these graphs.


\subsection{Designs}
\label{sec: designs - subgraphs}

For recent updates on the study of combinatorial block designs and orthogonal arrays (i.e., $t$-designs in the Johnson and Hamming graphs), we refer the reader to \cite{CD2007B}.
It should be remarked here that Keevash \cite{Keevash2014pre} has recently proved that, given $t$, $k$, and $\lambda$, the natural divisibility conditions for the existence of a block $t$-$(v,k,\lambda)$ design are also sufficient, provided that $v$ is large enough.
This generalizes the result of Wilson \cite{Wilson1975JCTA} for the case $t=2$ and that of Teirlinck \cite{Teirlinck1987DM} which establishes the existence of block $t$-designs for all $t$.

A number of simple $t$-designs over finite fields (i.e., $t$-designs in the Grassmann graphs) with $t$ at most $3$ have been constructed by many researchers; see, e.g., \cite{BKOW2013pre} and the references therein.
Recently, Fazeli, Lovett, and Vardy \cite{FLV2013pre} showed that non-trivial simple $t$-designs over finite fields $\mathbb{F}_q$ exist for all $t$ and $q$.

Delsarte $T$-designs in a distance-regular graph with $|T|=D-1$ (where $D$ is the diameter of the graph) have dual degree $1$.
Such designs are necessarily completely regular with covering radius $1$, and will be briefly discussed in Section \ref{sec: crc in other drg}.


\subsection{The Terwilliger algebra with respect to a code}\label{sec:Terwilliger algebra w.r.t. code}

Let $\G$ be a distance-regular graph with diameter $D\geq 3$, adjacency matrix $A$, and eigenvalues $k=\theta_0>\theta_1>\dots>\theta_D$, and let $C$ be a non-empty subset of $V_{\G}$ (i.e., a code) with covering radius $\rho$.
Let $\{C_0=C,C_1,\dots,C_{\rho}\}$ be the distance partition with respect to $C$, and let $\chi_i$ be the characteristic vector of $C_i$ ($i=0,1,\dots,\rho$).
For each $i=0,1,\dots,\rho$, let $E_i^{\ster}=E_i^{\ster}(C)$ be the diagonal matrix in $M_{v\times v}(\mathbb{C})$ with diagonal entries $(E_i^{\ster})_{yy}=(\chi_i)_y$.
The \emph{Terwilliger algebra} $\TT=\TT(C)$ \emph{with respect to} $C$ is the subalgebra of $M_{v\times v}(\mathbb{C})$ generated by $A,E_0^{\ster},E_1^{\ster},\dots,E_{\rho}^{\ster}$.
The algebra $\TT(C)$ was first introduced and studied by Martin and Taylor \cite{MT1997pre} for binary Hamming graphs.
We shall use the same terminology as in the case of the ordinary Terwilliger algebra (i.e., with respect to a vertex); cf.~Section \ref{sec:Talgebra}.
However, as observed by Martin and Taylor \cite{MT1997pre} and Suzuki \cite{Suzuki2005JAC}, the primary $\TT$-module is thin (and is therefore equal to $\mathrm{span}_{\mathbb{C}}\{\chi_0,\chi_1,\dots,\chi_{\rho}\}$) precisely when $C$ is a completely regular code.

Suzuki \cite{Suzuki2005JAC} studied irreducible $\TT$-modules in detail.
The results in \cite{Suzuki2005JAC} generalize (to some extent) both the theory of tight graphs (cf.~Go and Terwilliger \cite{GT2002EJC}) and the theory of the width of a code (cf.~Brouwer et al. \cite{BGKM03}).
Suzuki \cite{Suzuki2005JAC} showed that a $\TT(C)$-module with endpoint $\nu$ is also a $\TT(C_{\nu})$-module with endpoint $0$, where irreducibility and thinness are also preserved.
This allows us to focus on the irreducible modules with endpoint $0$.

Recall that the width of $C$ is defined by $w=\max\{i:\chi_0^{\mathsf{T}}A_i\chi_0\ne 0\}$, where $A_i$ is the distance-$i$ matrix of $\G$ ($i=0,1,\dots,D$); cf.~Section \ref{sec: posets}.
For $i=0,1,\dots,D$, let $E_i$ be the primitive idempotent associated with $\theta_i$.
Let $\mathbf{v}$ be a nonzero vector in $E_0^{\ster}\mathbb{C}^v$.
Then it is easy to see that there is a polynomial $f$ of degree at most $w$ such that $||E_i\mathbf{v}||^2=f(\theta_i)m(\theta_i)$ ($i=0,1,\dots,D$), where $m(\theta)$ denotes the multiplicity of an eigenvalue $\theta$ of $\G$.
This immediately gives the inequality $w\geq D-r(\mathbf{v})$, where $r(\mathbf{v})=|\{i:E_i\mathbf{v}\ne 0\}|-1$.
(Note that $r(\chi_0)$ is the dual degree of $C$.)
The vector $\mathbf{v}$ is said to be \emph{tight} (with respect to $C$) if $w=D-r(\mathbf{v})$.
Suzuki \cite{Suzuki2005JAC} showed among other results that if $\mathbf{v}$ is tight then $\TT\mathbf{v}$ is a thin irreducible $\TT$-module with endpoint $0$.
This result was previously obtained by Brouwer, Godsil, Koolen, and Martin \cite{BGKM03} for $\mathbf{v}=\chi_0$, and generalizes a theorem of Go and Terwilliger \cite[Thm.~9.8]{GT2002EJC}.
An important consequence is that if $\G$ is $Q$-polynomial then every irreducible module of the ordinary Terwilliger algebra $\TT(x)$ with \emph{displacement} $0$ is thin.\footnote{The \emph{displacement} (\cite{Terwilliger2005GC}) of an irreducible $\TT(x)$-module $W$ is $\eta=e+e^{\ster}+\delta-D$, where $e,e^{\ster},\delta$ are the endpoint, dual endpoint, and the diameter of $W$, respectively. It follows from Caughman's results \cite[Lemmas 5.1, 7.1]{Caughman1999DM} that $0\leq\eta\leq D$.}
This fact was used, e.g., to extend the Assmus-Mattson theorem; cf.~\cite[\S 5]{Tanaka2009EJC}.

Hosoya and Suzuki \cite{HS2007EJC} called $\G$ \emph{tight with respect to} $C$ if the orthogonal complement of $\mathrm{span}_{\mathbb{C}}\{\chi_0\}$ in $E_0^{\ster}\mathbb{C}^v$ is spanned by tight vectors.
By \cite[Thm.~13.6]{GT2002EJC}, $\G$ is tight in the sense of Section \ref{sec:tightDRG} if and only if $\G$ is non-bipartite and tight with respect to $\G_1(x)$ for some (or all) $x\in V_{\G}$.
Hosoya and Suzuki also introduced a homogeneity \emph{with respect to} $C$ in terms of the partition of $V_{\G}$ by the distances from both $C$ and a fixed vertex in $C$, and studied the relation between these two concepts.
They moreover showed that if $\G$ is $Q$-polynomial then the dual eigenmatrix of the association scheme induced on a descendent (cf.~Section \ref{sec: posets}) of $\G$ satisfies a certain system of linear equations, which in particular implies that $\G$ is tight with respect to every descendent.
This system of linear equations turned out to be fundamental to the study of descendents; cf.~\cite{Tanaka2009LAAb,Tanaka2011EJC}.
Lee \cite{Lee2013PhD} studied the above partition for Delsarte cliques (which are descendents of width $1$) in $Q$-polynomial distance-regular graphs that have the most general $q$-\emph{Racah type},\footnote{In the notation of Bannai and Ito \cite[\S III.5]{bi} (cf.~\cite{Talgebra92}), this $Q$-polynomial structure satisfies type I with $s\ne 0$ and $s^*\ne 0$. The polygons are the only known examples of this type.} and showed among other results that there is a natural action of the double affine Hecke algebra of type $(C_1^{\vee},C_1)$ on the subspace of $\mathbb{C}^v$ spanned by the characteristic vectors of the cells of the partition.
See \cite{TTW16+} for detailed information on the Terwilliger algebra with respect to a descendent.
