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

\subsection{Miscellaneous definitions}\label{sec:miscdefs}

A non-complete $k$-regular graph $\G$ on $v$ vertices is called
{\em strongly regular} with parameters $(v,k,\lambda,\mu)$ if each
two adjacent vertices have $\lambda$ common neighbors, and each two
nonadjacent vertices have $\mu$ common neighbors. Thus, the
connected strongly regular graphs are precisely the
distance-regular graphs with diameter two. The definition of an
{\em amply regular graph} with parameters $(v,k,\lambda,\mu)$ is
obtained by replacing the condition on the nonadjacent vertices by
the condition that each two vertices at distance $2$ have $\mu$
common neighbors.

For a graph $\G$ and $x \in V$, the graph induced on the set $\G_i(x)$
is called an $i$-th {\em subconstituent} of $\G$. The first subconstituent
in consideration is also called a {\em local graph} of $\G$, and is denoted
by $\Upsilon(x)$. We say that $\G$ is {\em locally} $\Delta$ if all local
graphs are isomorphic to $\Delta$. More generally, we let $\Upsilon(x,y)$ be
the induced subgraph on the set of common neighbors of $x$ and $y$ (so it is a
local graph of a local graph if $x$ and $y$ are adjacent), etc.. A {\em
Terwilliger graph} is a non-complete graph such that $\Upsilon(x,y)$ is a
clique of size $\mu$ for each two vertices $x$ and $y$ at distance two, for
some $\mu$. Thus, a Terwilliger graph has no induced quadrangles.


Let $\G$ be a distance-regular graph with valency $k$ and diameter $D$. Let
$\ell(c, a, b) = |\{i= 1,2, \dots, D-1 : (c_i, a_i, b_i) = (c,a,b)\}|$. In
particular, let $h = h(\G)$ and $t= t(\G)$ be defined by $h(\G) =
\ell(c_1, a_1, b_1)$ and $t(\G) = \ell(b_1,a_1, c_1) $. The parameter
$h(\G)$ is called the {\em head} of $\G$ and $t(\G)$ is called the
{\em tail} of $\G$.

The {\em girth} of $\G$ is the length of its shortest cycle. The
{\em numerical girth} of $\G$ is $2h+3$ if $c_{h+1} = 1$, else it
is $2h+2$. If $a_1 = 0$, then the girth is equal to the numerical
girth. If $\G$ is locally a disjoint union of cliques, then the
{\em geometric girth} of $\G$ is the minimal length of a cycle for
which the induced subgraph on each triple of its vertices is not a
triangle; this equals the numerical girth. If $\G$ has a local
graph that is not a disjoint union of cliques, then the geometric
girth is defined as $3$. For geometric graphs, the geometric girth
is half the girth of the incidence graph of the corresponding
partial linear space (see Section
\ref{sec:generalizations+geometric}). Note that the girth and the
numerical girth are determined by the intersection array, but in
general the geometric girth is not (for example, the Doob graphs
have geometric girth $3$, whereas the Hamming graphs (with the same
intersection array) have geometric girth $4$).

A quadruple $(x,y,z,u)$ of vertices is called a {\em parallelogram of length
$i$} if $d(x,y) = 1 = d(z,u)$, $d(x,z) = d(y,u) = d(y, z) = i-1$, and $d(x,u) =
i$. The graph $\G$ is called {\em $m$-parallelogram-free} for some
$m=2,3,\ldots, D$ if $\G$ does not contain any parallelogram of length at
most $m$. We say $\G$ is parallelogram-free if it does not contain any
parallelogram. Related conditions called $\mathrm{(CR)}_m$ and $\mathrm{(SS)}_m$ are given by Hiraki
\cite{Hi198, Hi499}.

A quadruple $(x,y,z,u)$ of vertices of $\G$ is called a \emph{kite of
length} $i$ if $d(x,y)=d(x,z)=d(y,z)=1$, $d(x,u)=i$, and $d(y,u)=d(z,u)=i-1$.

A subgraph $\Delta$ of $\G$ is called \emph{geodetically closed}, or \emph{closed} for short,  if $z \in
V_{\Delta}$ for all $x,y \in V_{\Delta} $ and $z$ on a geodetic between $x$ and $y$.  (A closed subgraph is also called
{\em convex} by some authors.) The subgraph $\Delta$ is called {\em strongly closed} if $z \in V_{\Delta}$ for all
vertices $x, y \in V_{\Delta}$ and $z \in V_{\G}$ such that $d_{\G}(x,z) + d_{\G}(z,y) \leq d_{\G}(x,y)
+ 1$. (The term \emph{weak-geodetically closed} is also used for strongly closed.) It is known that if $c_2>1$ then all
strongly closed subgraphs are regular; cf.~\cite[Lemma~5.2]{W98} or \cite{Su295}. A distance-regular graph $\G$
with diameter $D$ is said to be $m$-\emph{bounded} for some $m=1,2, \ldots, D$ if for all $i=1,2, \ldots, m$ and all
vertices $x$ and $y$ at distance $i$ there exists a strongly-closed subgraph $\Delta(x,y)$ with diameter $i$,
containing $x$ and $y$ as vertices. (Note that Weng \cite{W97,We99} also required that $\Delta(x,y)$ is regular.)


\subsection{A few comments on the eigenspaces}

Consider an association scheme with primitive idempotents $E_0,E_1,\dots,E_D$.
By computing the squared norm, it follows that
\begin{equation}\label{3-tensor}
	\sum_{x\in V}E_i\mathbf{e}_x\otimes E_j\mathbf{e}_x\otimes E_h\mathbf{e}_x=0 \quad \text{if and only if}\ q_{ij}^h=0 \quad (h,i,j=0,1,\ldots, D),
\end{equation}
where $\mathbf{e}_x\in\R^v$ denotes the characteristic vector of $\{x\}$.
In fact, this computation also gives an alternative proof of the Krein conditions; cf.~Proposition \ref{Krein condisions}.
Recall that the absolute bound (cf.~Proposition \ref{absolute bound}) was an immediate consequence of the obvious observation that $(E_i \circ E_j) \R^v \subseteq \mathrm{span} (E_i \R^v \circ E_j \R^v)$.
We remark here that these two subspaces indeed coincide:
\begin{equation}\label{colsp Hadamard}
	\mathrm{span} (E_i \R^v \circ E_j \R^v) = (E_i \circ E_j) \R^v = \sum_{q_{ij}^h \ne
0} E_h \R^v  \quad (i,j=0,1,\dots,D).
\end{equation}
To see this, note that $\langle \mathbf{u}\circ\mathbf{v},\mathbf{w}\rangle=\langle \mathbf{u}\otimes\mathbf{v}\otimes\mathbf{w},\sum_{x\in V}E_i\mathbf{e}_x\otimes E_j\mathbf{e}_x\otimes E_h\mathbf{e}_x\rangle$ for all $\mathbf{u}\in E_i\R^v$, $\mathbf{v}\in E_j\R^v$, and $\mathbf{w}\in E_h\R^v$, where $\langle\, ,\, \rangle$ denotes the standard inner product.
Therefore, it follows from \eqref{3-tensor} that $\mathrm{span} (E_i \R^v \circ E_j \R^v)$ is orthogonal to $E_h\R^v$ whenever $q_{ij}^h=0$.
These results are due to Cameron, Goethals, and Seidel \cite{CGS1978NAWIM} (cf.~\cite[\S II.8]{bi}, \cite{Tanaka2009LAAa}), and are quite fundamental in the theory of distance-regular graphs and association schemes; see, e.g., Sections \ref{sec:Talgebra} and \ref{sec:triple intersection numbers}.
We note that, in view of \eqref{colsp Hadamard}, the ordering $E_0,E_1,\dots,E_D$ is $Q$-polynomial if and only if $\sum_{\ell=0}^i (E_1\R^v)^{\circ\ell}=\sum_{\ell=0}^i E_{\ell}\R^v$ for all $i=0,1,\dots,D$, where $(E_1\R^v)^{\circ\ell}=E_1\R^v\circ E_1\R^v\circ\dots\circ E_1\R^v$ ($\ell$ times).


\subsection{The Terwilliger algebra}\label{sec:Talgebra}

The \emph{Terwilliger} (or \emph{subconstituent}) \emph{algebra} of an association scheme was introduced in
\cite{Talgebra92}. Though it should be stressed that this algebra also plays an important role in the theory of general
distance-regular graphs (cf.~Section \ref{sec:Talg6}), it is particularly well-suited for $Q$-polynomial
distance-regular graphs. In fact, this algebra has (part of) its roots in the study of \emph{balanced sets}
(cf.~\eqref{balanced set condition}); see, e.g., \cite[p.~93,~Note 1]{Terwilliger1988GC}.

In the context of the Terwilliger algebra, the Bose-Mesner algebra of an association scheme is always assumed to be
\emph{over} $\mathbb{C}$, that is,
\begin{equation*}
	\AL=\mathrm{span}_{\mathbb{C}}\{A_0,A_1,\dots,A_D\}\subset M_{v\times v}(\mathbb{C}).
\end{equation*}
Fix a `base vertex' $x\in V$. For each $i=0,1,\dots,D$, let $E_i^{\ster}=E_i^{\ster}(x)$, $A_i^{\ster}=A_i^{\ster}(x)$
be the diagonal matrices\footnote{We use $\ster$-notation instead of the usual $\ast$-notation in order to
avoid confusion with the conjugate transpose.} in $M_{v\times v}(\mathbb{C})$ with diagonal entries $(E_i^{\ster})_{yy}=(A_i)_{xy}$,
$(A_i^{\ster})_{yy}=v(E_i)_{xy}$.
Note that $E_i^{\ster}E_j^{\ster}=\delta_{ij}E_i^{\ster}$,
$\sum_{i=0}^DE_i^{\ster}=I$, and moreover $A_i^{\ster}A_j^{\ster}=\sum_{h=0}^Dq_{ij}^hA_h^{\ster}$. These matrices span
the \emph{dual Bose-Mesner algebra} $\AL^{\ster}=\AL^{\ster}(x)$ \emph{with respect to} $x$:
\begin{equation*}
	 \AL^{\ster}=\mathrm{span}_{\mathbb{C}}\{E_0^{\ster},E_1^{\ster},\dots,E_D^{\ster}\}=\mathrm{span}_{\mathbb{C}}\{A_0^{\ster},A_1^{\ster},\dots,A_D^{\ster}\}\subset M_{v\times v}(\mathbb{C}).
\end{equation*}
Note that if the association scheme is $Q$-polynomial with respect to the ordering
$(E_i)_{i=0}^D$ then $A_1^{\ster}$ generates $\AL^{\ster}$. The \emph{Terwilliger
algebra} $\TT=\TT(x)$ \emph{with respect to} $x$ is the
subalgebra of $M_{v\times v}(\mathbb{C})$ generated by $\AL$ and $\AL^{\ster}$
\cite{Talgebra92}. The following are relations in $\TT$:
\begin{equation}\label{relations}
	E_i^{\ster}A_jE_h^{\ster}=0 \ \Leftrightarrow p_{ij}^h=0, \quad E_iA_j^{\ster}E_h=0 \ \Leftrightarrow q_{ij}^h=0 \quad (h,i,j=0,1,\ldots, D).
\end{equation}
We note that the latter is a variation of \eqref{3-tensor}. Because $\TT$ is closed under conjugate-transposition, it
is semisimple and every two non-isomorphic irreducible $\TT$-modules in $\mathbb{C}^v$ are orthogonal.
Let $G$ be the full automorphism group of the association scheme.
Then $\TT$ is a subalgebra of the centralizer algebra\footnote{Dunkl \cite{Dunkl1976IUMJ,Dunkl1978SIAM,Dunkl1978MM,Dunkl1979P} and Stanton \cite{Stanton1981GD} studied this latter algebra in detail in the context of addition theorems for orthogonal polynomials associated with some classical families of distance-regular graphs.} of the action
of the stabilizer $G_x$ of $x$ on $\mathbb{C}^v$.
The two algebras are known to be equal, e.g., for Hamming graphs;
cf.~\cite[Prop.~3]{GST2006JCTA}. We also note that the structure of $\TT$ may depend on the choice of $x$ if $G$ is not
transitive on $V$; cf.~Section \ref{twistedsection}.

Let $W$ be an irreducible $\TT$-module. When the association scheme is $P$-polynomial (resp.~$Q$-polynomial) with
respect to the ordering $(A_i)_{i=0}^D$ (resp.~$(E_i)_{i=0}^D$), we define the \emph{endpoint} (resp.~\emph{dual
endpoint}) of $W$ by $\min\{i\, :\, E_i^{\ster}W\ne 0\}$ (resp.~$\min\{i\, :\, E_iW\ne 0\}$). We call $W$ \emph{thin}
(resp.~\emph{dual thin}) if $\dim E_i^{\ster}W\leq 1$ (resp.~$\dim E_iW\leq 1$) for $i=0,1,\dots,D$. We also define the
\emph{diameter} and the \emph{dual diameter} of $W$ by $|\{i\, :\, E_i^{\ster}W\ne 0\}|-1$ and $|\{i\, :\, E_iW\ne
0\}|-1$, respectively. If the association scheme is $P$-polynomial
(resp.~$Q$-polynomial), then thin (resp.~dual thin) implies dual
thin (resp.~thin) \cite{Talgebra92}. There is a unique irreducible
$\TT$-module with $E_0^{\ster}W\ne 0$ and $E_0W\ne 0$, called the
\emph{primary} (or \emph{trivial}) $\TT$-module; it is thin, dual
thin, and given by
$\mathrm{span}_{\mathbb{C}}\{A_0\mathbf{e}_x,A_1\mathbf{e}_x,\dots,A_D\mathbf{e}_x\}$,
where $\mathbf{e}_x\in\mathbb{C}^v$ denotes the characteristic
vector of $\{x\}$. We say the association scheme is $i$-\emph{thin
with respect to} $x$ if every irreducible $\TT(x)$-module $W$ with
$E_i^{\ster}W\ne 0$ is thin.\footnote{The definition of the
$i$-thin condition here is taken from
\cite{Dickie1995D,DT1998JAC}. This is slightly different from the
standard one for the case when the association scheme is
$P$-polynomial, where it is called $i$-\emph{thin with respect to}
$x$ if every irreducible $\TT(x)$-module with endpoint at most $i$
is thin. On the other hand, the present definition of course has
the advantage that it makes sense for general association schemes.
We shall be careful below not to cause any confusion when we
discuss results involving this concept.} It is said to be
\emph{thin with respect to} $x$ if it is $i$-thin with respect to
$x$ for all $i=0,1,\dots,D$. Finally, we say the association scheme
is \emph{thin} (resp.~$i$-\emph{thin}) if it is thin
(resp.~$i$-thin) with respect to $x$ for all $x\in V$.

In the study of the Terwilliger algebra, it is often quite important to consider the following three matrices:
\begin{equation}\label{quantum decomposition}
	L=\sum_{i=1}^DE_{i-1}^{\ster}AE_i^{\ster}, \quad F=\sum_{i=0}^DE_i^{\ster}AE_i^{\ster}, \quad R=\sum_{i=0}^{D-1}E_{i+1}^{\ster}AE_i^{\ster},
\end{equation}
called the \emph{lowering}, \emph{flat}, and \emph{raising matrices},
respectively. Note that $A=L+F+R$. As an illustrative example, suppose $\G$
is the $D$-cube $H(D,2)$, and let
$A^{\ster}=A_1^{\ster}=\sum_{i=0}^D(D-2i)E_i^{\ster}$ correspond to the
$Q$-polynomial idempotent $E_1$ associated with the second largest eigenvalue
$\theta_1=D-2$. Then $F=0$ because $\G$ is bipartite, and it follows that
$L,R$, and $A^{\ster}$ generate $\TT$. Moreover, we can easily verify that
$LR-RL=A^{\ster}$, $RA^{\ster}-A^{\ster}R=2R$, and $LA^{\ster}-A^{\ster}L=-2L$,
so that the Terwilliger algebra $\TT$ is a homomorphic image of the universal
enveloping algebra of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. Therefore,
every irreducible $\TT$-module $W$ has the structure of an irreducible
$\mathfrak{sl}_2(\mathbb{C})$-module, and $\bigoplus_{i=e}^{D-e}E_i^{\ster}W$
gives the weight space decomposition of $W$, where $e$ denotes the endpoint of
$W$. In particular, $H(D,2)$ is thin. We refer the reader to Terwilliger
\cite{Terwilliger1993N} and Go \cite{Go2002EJC} for more details.

\subsection{Equitable partitions and completely regular codes}\label{sec:CRC}

\subsubsection{Interlacing, the quotient matrix, and the quotient
graph}\label{sec:interlacing}

Eigenvalue interlacing is a useful tool in studying distance-regular graphs,
and more generally, in spectral graph theory; see the survey by Haemers
\cite{HaeInterlacing}. A sequence of numbers $\mu_1 \geq \mu_2 \geq \cdots \geq
\mu_m$ is said to {\em interlace} a sequence $\lambda_1 \geq \lambda_2 \geq
\cdots \geq \lambda_n$, with $n>m$, if $\lambda_i \geq \mu_i \geq
\lambda_{n-m+i}$ for all $i=1,2,\dots,m$. The interlacing is called {\em tight}
if for some $k \in \{0,1,\dots,m\}$ the equalities $\lambda_i=\mu_i,
i=1,\dots,k$ and $\lambda_{n-m+i}=\mu_i, i=k+1,\dots,m$ hold.

An elementary interlacing result states that the eigenvalues of a principal
submatrix $B$ of a symmetric matrix $A$ interlace the eigenvalues of $A$
itself. When applied to graphs: the eigenvalues of an induced subgraph of a
graph $\G$ interlace the eigenvalues of $\G$.

A somewhat more complicated --- but very useful --- result concerns the
so-called quotient matrix. Let $\Pi = \{P_1, P_2, \ldots, P_t\}$ be a partition
of the vertex set of a graph $\G$. Let $f_{ij}$ be the average number of
neighbors in $P_j$ of a vertex in $P_i$, for $i,j =1,2, \ldots, t$. The matrix
$F = (f_{ij})$ is called the {\em quotient matrix} of $\Pi$. The partition
$\Pi$ is called {\em equitable} if every vertex in $P_i$ has exactly $f_{ij}$
neighbors in $P_j$. Also the eigenvalues of $F$ interlace the eigenvalues of
$\G$. Moreover, if the interlacing is tight, then the partition is
equitable. In this case, it can easily be seen that an eigenvector ${\bf u}$ of
$F$ can be `blown up' to an eigenvector ${\bf v}$ of $\G$ (with the same
eigenvalue) by setting $v_x = u_i$ if $x \in P_i$. An example of an equitable
partition in a distance-regular graph $\G$ is the distance partition $\Pi
=\{\G_0(z),\G_1(z),\dots,\G_D(z)\}$ of a vertex $z$, and its
quotient matrix is the intersection matrix $L$ as in (\ref{matrixl}).

Given a partition $\Pi= \{ P_1, P_2, \ldots, P_t\}$ of the vertex set of a
graph $\G$, we define the {\em quotient graph} $\G/ \Pi$ with vertex
set $\Pi$, where $P_i \sim P_j$ if $i \neq j$ and there  exist $x \in P_i$ and
$y \in P_j$ such that $x \sim y$ in $\G$.

We call an equitable partition $\Pi$ {\em uniformly regular} if its quotient
matrix $F$ and the adjacency matrix $B$ of $\G/ \Pi$ are related as
$F=fI+\tilde{f}B$, for some numbers $f$ and $\tilde{f} \neq 0$. It is clear
that in this case, the eigenvalues of the quotient $\G/ \Pi$ follow in a
straightforward way from the eigenvalues of $F$, and the latter are eigenvalues
of $\G$, as we just observed. An example of a uniformly regular partition
is given by the partition into fibres of an antipodal distance-regular graph.
In this case, the corresponding quotient graph is the folded graph.

\subsubsection{Completely regular codes}\label{sec:crcdelsarte}

Let $\G$ be a connected graph, say with diameter $D$, and let $C$ be a subset of $V=V_{\G}$. For $i \geq 0$,
let $C_i = \{ x \in V : d(x, C) = i\}$, where $d(x, C) = \min\{ d(x,c) : c \in C\}$. The {\em covering radius} of $C$,
denoted by $\rho=\rho(C)$, is the maximum $i$ such that $C_i \neq \emptyset.$ The subset (or code) $C$ is called {\em
completely regular} if the distance partition $\Pi = \{C_i : i=0,1, \ldots, \rho\}$ is equitable. Note that the
corresponding quotient matrix is tridiagonal; it is therefore common to denote $f_{i, i-1}, f_{i,i}$ and $f_{i,i+1}$ by
$\gamma_i, \alpha_i$, and $\beta_i$, respectively. These numbers are called the {\em intersection numbers} of $C$. This
definition of a completely regular subset (or code) was introduced by Neumaier \cite{Neu92} and he showed that for
distance-regular graphs it is equivalent to Delsarte's definition \cite[p.~67]{del} in terms of the so-called outer
distribution. It is clear that if $C$ is completely regular then so is $C_{\rho}$. Note that for a distance-regular
graph, every singleton $\{z\}$ is a completely regular code with $ \gamma_i= c_i$, $\alpha_i= a_i$, and $\beta_i =
b_i$. In general, the behavior of the intersection numbers of a completely regular code can however be quite different
from that of the intersection numbers of a distance-regular graph. For example, it is not true in general that the
$\gamma_i$ are non-decreasing; see \cite{Ko95}. For more background information on completely regular codes, we refer
to the work of Martin \cite{Martinthesis, MartinCRC}.

A partition $\Pi = \{ P_1, P_2, \ldots, P_t\}$ of $V$ is called a {\em
completely regular partition} if it is equitable and all of the $P_i$ are
completely regular with the same intersection numbers. It is known that a
completely regular partition is uniformly regular. A typical (and motivating)
example of a completely regular partition is the partition into cosets of a
linear code $C$ of length $n$ over $GF(q)$ that is completely regular in the
Hamming graph $H(n,q)$.
(More generally, we can consider a completely regular additive code in a translation distance-regular graph.\footnote{A \emph{translation} distance-regular graph is a distance-regular Cayley graph on an abelian group. An \emph{additive} code in such a graph is just a subgroup of the abelian group ($=$ vertex set).})
In this case we call $\G/ \Pi$ the coset graph of
$C$. This coset graph is distance-regular by the following result.


\begin{theorem}\label{uniformlyregularpartition}
{\em \cite[Thm.~11.1.5,~11.1.8]{bcn}} Let $\G$ be a distance-regular graph and $\Pi$ a uniformly regular partition
of $\G$ with quotient matrix $F$. Then $\G/ \Pi$ is distance-regular if and only if $\Pi$ is completely
regular. Moreover, if so, then the intersection numbers of $\G/ \Pi$ can be explicitly calculated from the
intersection numbers of $\G$ and $F$.
\end{theorem}


%\subsubsection{Delsarte cliques}\label{sec:Delsartecliques}

%Delsarte \cite[p.~31]{del} obtained a linear programming bound for cliques in strongly regular graphs. It was observed
%by Godsil \cite[p.~276]{Godsilac} that the same bound holds for distance-regular graphs, as follows.
%
%\begin{prop}\label{delbound} Let $\G$ be a distance-regular graph with valency $k$ and smallest
%eigenvalue $\theta_{\min}$. Let $C$ be a clique in $\G$ with $c$ vertices.
%Then $c \leq 1 - \frac{k}{\theta_{\min}}.$
%\end{prop}
%
%\begin{proof}
%Let $\chi$ be the characteristic vector of $C$, and let $E$ be the
%primitive idempotent corresponding to $\theta_{\min}$. The result follows from
%working out $\chi^{\top}E\chi \geq 0$.
%\end{proof}
%
%\noindent A clique $C$ in a distance-regular graph $\G$ that attains the
%above bound is called a {\em Delsarte clique}. The following result
%characterizes such cliques.

Delsarte cliques are examples of completely regular codes. Indeed,
the following result characterizes such cliques.

\begin{prop} {\em \cite[Lemmas~13.7.2,~13.7.4]{Godsilac}}
Let $\G$ be a distance-regular graph with valency $k$, diameter $D$ and
smallest eigenvalue $\theta_{\min}$. Let $C$ be a clique in $\G$ with $c$
vertices. Then $C$ is a Delsarte clique if and only if $C$ is a completely
regular code with covering radius $D-1$. Moreover, if so, then $\phi_i u_i +
(c-\phi_i)u_{i+1} = 0$, where $(u_i)_{i=0}^D$ is the standard sequence for
$\theta_{\min}$ and $\phi_i= |\G_i(x) \cap C| $ for a vertex $x$ at
distance $i$ from $C$.
\end{prop}

\noindent Note that the equation $\phi_i u_i + (c-\phi_i)u_{i+1} = 0$ follows from the fact that $E\chi = 0$ (with $E$
and $\chi$ as in the proof of Proposition \ref{delbound}).
Indeed, if $E=UU^{\top}$, then $U^{\top}\chi=0$, and hence for
the corresponding representation associated to $\theta_{\min}$ (see
Section
\ref{sec2:evmult}) we have that
\begin{equation}\label{eq:repdelsarte}
\sum_{z\in C}\hat{z}=0.
\end{equation}
Taking the inner product with $\hat{x}$, where $x$ is a vertex at
distance $i$ from $C$ gives the required equation.
 This implies (by using
\cite[Thm.~4.1]{Neu92}) that the intersection numbers of a Delsarte
clique can be explicitly calculated from the intersection numbers
of $\G$.

For a subset of the vertex set of an association scheme, with characteristic vector $\chi$, the {\em degree} and {\em
dual degree} are defined by $|\{i \neq 0:\chi^{\mathsf{T}}A_i\chi\ne 0\}|$ and $|\{i \neq 0:\chi^{\textsf{T}}E_i\chi\ne
0\}|$, respectively.


\subsection{Distance-biregular graphs and weakly geometric graphs}\label{sec:generalizations+geometric}

For an arbitrary graph with vertices $x$ and $y$ at distance $i$, we define
$c_i(x,y)$, $a_i(x,y)$, and $b_i(x,y)$ as the numbers of neighbors of $y$ that
are at distance $i-1$, $i$, and $i+1$, respectively. Thus, a connected graph
with diameter $D$ is distance-regular if these numbers do not depend on $x$ and
$y$ (but only on their distance $i$). If in an arbitrary graph the numbers
$c_i(x,y)$, $a_i(x,y)$, or $b_i(x,y)$ do not depend on $x$ and $y$, for some
$i$, then we will write $c_i$, $a_i$, or $b_i$, respectively (as in
distance-regular graphs). For example, in an arbitrary bipartite graph, one has
$a_i=0$ for all $i$.

For ease of notation and formulation, we will call the two biparts of a
bipartite graph its color classes $R$ and $B$, and say that a vertex in $R$ is
red, and a vertex in $B$ is blue.

Now a connected bipartite graph is called {\em distance-biregular} if the
numbers $c_i(x,y)$ and $b_i(x,y)$ depend only on $i$ and the color of $x$. We
denote these numbers by $c_i^R$, $b_i^R$, $c_i^B$, and $b_i^B$. Straightforward
examples are the complete bipartite graphs.

We say that a graph $\G$ is distance-regular around a vertex $x$ if the
singleton $\{x\}$ is a completely regular code in $\G$. A well-known result
by Godsil and Shawe-Taylor \cite{GSbiregular} states that if $\G$ is a
connected graph that is distance-regular around every vertex, then $\G$ is
distance-regular or distance-biregular.

A bipartite graph is called {\em semiregular} (or biregular) if the valency of
a vertex only depends on its color. We denote these valencies by $k_R$ and
$k_B$.

Powers \cite{Powerssemiregular} used the term semiregular for a concept that he
introduced, and what we now call distance-semiregular (following Suzuki
\cite{Su95, Su99}). A connected bipartite graph is called {\em
distance-semiregular} with respect to one of its color classes, $R$ say, if it
is distance-regular around all red vertices, with the same parameters (i.e,
there are $b_i^R$ and $c_i^R$ such that $b_i(x,y) = b_i^R$ and $c_i(x,y) =
c_i^R$ if $x \in R$ and $d(x,y) = i$). Note that every distance-biregular graph
is distance-semiregular, and in turn, each distance-semiregular graph is
semiregular, with valencies $k_R=b_0^R$ and $k_B=1+b_1^R$. The Hoffman graph
\cite{hoffman63} (the unique graph cospectral but not isomorphic to $H(4,2)$)
is an example of a (regular!) distance-semiregular graph that is not
distance-biregular.

Let $\G$ be distance-semiregular with respect to $R$, then its halved graph
$\G^R_2$ (i.e., the distance-2 graph of $\G$, induced on $R$) is
distance-regular. Let $C = \G(x)$ for some blue vertex $x$. Then $C$ is a
clique in $\G^R_2$, that is also a completely regular code in $\G^R_2$.
This leads to the following definition.

A distance-regular graph $\Delta$ is called {\em weakly geometric}
(with respect to ${\cal C}$) if it contains a collection ${\cal C}$
of cliques such that each edge is contained in a unique $C \in
{\cal C}$ and all $C \in {\cal C}$ are completely regular codes
with the same parameters. Thus, a geometric distance-regular graph
(see Section \ref{sec:geometricgraphs}) is weakly geometric.
Because of the property that each edge is contained in a unique
clique, there is a naturally associated partial linear space, whose
points are the vertices of $\Delta$ and whose lines are the cliques
of ${\cal C}$, and incidence is defined by containment. The point
(or collinearity) graph of this partial linear space is $\Delta$.
The bipartite (point-line) incidence graph of the partial linear
space is a distance-semiregular graph with girth at least 6; in
fact, this gives a one-to-one correspondence between the latter
type of graphs and weakly geometric distance-regular graphs. The
partial linear space has also been studied by De Clerck, De Winter,
Kuijken, and Tonesi
\cite{DeDeKuTo06, KuiTon05} under the name {\em distance-regular
geometry}.




Using the same correspondence, certain distance-semiregular graphs
with girth 4 correspond to Delsarte graphs and Delsarte clique
graphs as introduced by Bang, Hiraki, and Koolen \cite{BaHiKo07}
(see also \cite{BaHiKo10}). Delsarte graphs and Delsarte clique
graphs are closely related to the geometric distance-regular graphs
of Section \ref{sec:geometricgraphs}.



 We remark that the Johnson graphs
$J(n,D)$ and Grassmann graphs $J_q(n,D)$ are not just (weakly)
geometric with respect to a set of Delsarte cliques (the
$(D-1)$-sets or $(D-1)$-dimensional subspaces; that is, the set of
vertices containing a fixed $(D-1)$-set or $(D-1)$-space is a
Delsarte clique), but also weakly geometric with respect to another
set of cliques, namely the $(D+1)$-sets or $(D+1)$-dimensional
subspaces (i.e., the sets of vertices contained in these),
respectively. The corresponding incidence graphs are
distance-biregular; for $n=2D+1$, we obtain the distance-regular
Doubled Odd graph and Doubled Grassmann graph, respectively. In
Section
\ref{sec:Geometricdrg} we will discuss geometric distance-regular
graphs in more detail.

Following Suzuki \cite{Su99}, we say a distance-regular graph $\G$
is {\em of order $(s, t)$} (for some integers $s$ and $t$) if it is
locally the disjoint union of $t+1$ cliques of size $s$. This is
equivalent to the property that $\G$ contains no induced complete
tripartite graph $K_{2,1,1}$ (a kite of length $2$).

A distance-regular graph $\G$ of order $(s,t)$ with diameter $D$ is called
a {\em regular near polygon} if $a_i = c_i a_1$ for all $i=1,2,\ldots,D-1$. If
$a_d=c_D a_1$ we call $\G$ a regular near $2D$-gon; otherwise it is called
a regular near $(2D+1)$-gon. A regular near polygon of diameter $D$ is
geometric if and only if it is a regular near $2D$-gon.  We say $\G$ is
{\em thick} if $s \geq 2$ (the regular near polygons with $s=1$ are exactly the
bipartite distance-regular graphs and the generalized odd graphs).

Weng \cite{We99} defined a distance-regular graph to have {\em
geometric parameters $(D, b, \alpha)$} if it has classical
parameters $(D, b, \alpha, \beta)$ with $b \neq 1$ and $\beta =
\alpha\frac{1+b^D}{1-b}$. He used this concept in the partial
classification of distance-regular graphs with classical parameters
with $b<-1$. This does not seem to be related to geometric
distance-regular graphs.



\subsection{Homogeneity}\label{sec:homogeneity4}




Let $\G$ be a connected graph. For two distinct vertices $x$ and $y$,
define $\G_{i,j}(x,y) = \G_i(x) \cap \G_j(y)$. If it is clear (or
irrelevant) which pair $x,y$ is meant we will write $\G_{i,j}$ instead of
$\G_{i,j}(x,y)$. For $u \in \G_{i,j}$, let $p_{i,j;r,s}(u) = |\{ z \in
\G_{r,s} : z \sim u\}|$. We say the parameter $p_{i,j;r,s}$ exists with respect
to the pair $x,y$ if $p_{i,j;r,s}(u) =p_{i,j;r,s}(u')$ for all $u, u' \in
\G_{i,j}(x,y)$.


A connected graph $\G$ with diameter $D$ is called {\em $i$-homogeneous}
(in the sense of Nomura), $i =0,1, \ldots, D$ if for all pairs $x,y$ at
distance $i$ and all $r,s,r', s' \in \{0,1, \ldots, D\}$, the parameter
$p_{r,s; r', s'}$ exists and does not depend on the pair $x,y$, or in other
words, the partition $\{\G_{i,j}(x,y) : \G_{i,j}(x,y) \neq \emptyset, i,j =0,1,
\ldots, D\}$ is equitable for each pair $x,y$ at distance $i$ and the
parameters do not depend on the pair $x,y$.\footnote{In most of the literature
$\G_{i,j}$ is denoted as $D_i^j$. We chose different notation because $D$
stands for the diameter, and the superscript-subscript notation seems useful
only in intersection diagrams.}

Note that a $0$-homogeneous graph is distance-regular, and a $1$-homogeneous
graph is distance-regular. Examples of $1$-homogeneous distance-regular graphs
are the Johnson graphs $J(2D,D)$, the bipartite distance-regular graphs, and
the regular near $2D$-gons. To study $i$-homogeneous graphs, it is sometimes
useful to draw intersection diagrams with respect to two vertices $x$ and $y$.
In Figure \ref{pic:intersectiondiagram} we have an example of such an
intersection diagram for the Johnson graph $J(6,3)$.

\begin{figure}[h!]
\centering
\includegraphics[width=45mm,viewport=140 90 850 718,clip]{intersection-diagram.pdf}
\caption{Intersection diagram of $J(6,3)$} \label{pic:intersectiondiagram}
\end{figure}


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

Consider an association scheme with primitive idempotents $E_i$ ($i=0,1,\dots,D$).
Let $T$ be a subset of $\{1,2,\dots,D\}$.
A set $Y$ of vertices of the association scheme with characteristic vector $\chi$ is called a (\emph{Delsarte}) $T$-\emph{design} if $E_i\chi=0$ for all $i\in T$.
This definition is due to Delsarte \cite{del}.

Suppose that the association scheme is $Q$-polynomial with respect to the ordering $E_0,E_1,\dots,E_D$.
In this case, a $\{1,2,\dots,t\}$-design is simply called a $t$-design.
The \emph{strength} of $Y$ is then defined by $\min\{i\ne 0:E_i\chi\ne 0\}-1$, i.e., it is the maximum integer $t$ for which $Y$ is a $t$-design.
Delsarte \cite{del} showed that the $t$-designs in the Johnson graphs and Hamming graphs are precisely the combinatorial block $t$-designs and the orthogonal arrays of strength $t$, respectively.
A similar interpretation was established for the other classical families of distance-regular graphs by Delsarte \cite{Delsarte1976JCTA}, Munemasa \cite{Munemasa1986GC}, and Stanton \cite{Stanton1986GC}.

For more results on $T$-designs in association schemes, see the recent survey by Martin and Tanaka \cite{martintanaka} and the references therein.
