%Doubled Odd capitalized




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

In this section, we collect (relatively new) results on $Q$-polynomial
distance-regular graphs. Throughout this section, we shall use the following
notation unless otherwise stated. Let $\G$ denote a distance-regular graph
with diameter $D\geq 3$ and valency $k\geq 3$. Let $\theta$ be an eigenvalue of
$\G$, $E$ the corresponding primitive idempotent, and $(u_i)_{i=0}^D$ the
standard sequence with respect to $\theta$.

Suppose for the moment that $E$ is $Q$-polynomial, and let $E_0,E_1=E,E_2,\dots,E_D$ be the corresponding $Q$-polynomial ordering.
Then by Leonard's theorem
(cf.~\cite[\S 8.1]{bcn}) there exist $p,r,r^{\ster}\in\mathbb{C}$ such that
\begin{equation}\label{TTR}
	u_{i-1}+u_{i+1}=pu_i+r, \quad \theta_{i-1}+\theta_{i+1}=p\theta_i+r^{\ster} \quad (i=1,2,\dots,D-1).
\end{equation}
It should be remarked that the sequence of polynomials $(v_i)_{i=0}^D$ (see~\eqref{distancepolynomials}) belongs to the
terminating branch of the Askey scheme \cite{KS1998R,KLS2010B} of (basic) hypergeometric orthogonal polynomials. (We also allow the specialization\footnote{The polynomials corresponding to the case $q=-1$ have recently been receiving considerable attention; see, e.g., \cite{GVZ2014SIGMA} and the references therein.} $q\rightarrow -1$.)
See also \cite{Terwilliger2006N}.
We call $E$ \emph{classical} if $(u_{i-1}-u_i)/(u_i-u_{i+1})$
is independent of $i=1,2,\dots,D-1$. It follows that $\G$ has a classical $Q$-polynomial idempotent if and only if it
has classical parameters such that $p=b+b^{-1}$; cf.~\cite[Thm.~8.4.1]{bcn}, \cite[Prop.~6.2]{Tanaka2011EJC}.

We begin with discussions on graphs with classical parameters.



\subsection{The graphs with classical parameters with \texorpdfstring{$b=1$}{b=1}}
\label{sec: the case b=1}

All graphs with classical parameters with $b=1$ have been determined: the Hamming graphs, Doob graphs, halved cubes,
Johnson graphs, and the Gosset graph; cf.~\cite[Thm.~6.1.1]{bcn} or \cite{Neu-b1}. Main contributors to this
classification were Egawa \cite{E-b1}, who characterized the Hamming and Doob graphs, and Terwilliger \cite{Ter-b1,
Ter-root} and Neumaier \cite{Neu-b1}, who used the classification of root lattices and the representation with respect
to the second largest eigenvalue to come to the final classification.
We note that $b=1$ implies $\theta_1=b_1-1$,
and the graphs satisfying the latter have been classified; cf.~\cite[Thm.~4.4.11]{bcn}.
In \cite{KS94,Shp98}, Koolen and Shpectorov used metric theory to
classify the distance-regular graphs whose distance-matrix has exactly one
positive eigenvalue. The distance-regular graphs with classical parameters with
$b=1$ have this property.
Godsil \cite{G98} considered the convex hull of the representation with respect
to a fixed eigenvalue. He classified when the $1$-skeleton of this polytope with
respect to the second largest eigenvalue is isomorphic to the original
distance-regular graph. In his classification he again finds all
distance-regular graphs with classical parameters with $b=1$.



\subsection{Recent results on graphs with classical parameters}\label{sec:recentclassical}

Metsch \cite[Cor. 1.3]{Me99} showed that if $\G$ has classical parameters and is not a Johnson, Grassmann, Hamming, or bilinear forms graph, then the parameter $\beta$ is bounded in terms of $D, b,$ and $\alpha$.

Terwilliger \cite{Terwilliger1995EJC} showed that if $\G$ has classical parameters with $b<-1$ then $\G$ has no kites of any length $i=2,3,\dots,D$. This result,
combined with earlier work of Ivanov and Shpectorov \cite{IS1991JMSJ}, proves that the Hermitian forms graphs
$\mathit{Her}(D,q^2)$ with $D\ge 3$ are uniquely determined by their intersection arrays. See also \cite{Weng1995GC}. A
related result by Weng \cite{W98} is as follows (cf.~Section \ref{sec: strongly closed subgraphs}).

\begin{prop}
Suppose $\G$ is $Q$-polynomial with $D\ge 3$, $c_2>1$, and $a_1\ne 0$. Then
the following are equivalent:
\begin{enumerate}[{\em (i)}]
\item $\G$ has classical parameters, and either $b < -1$, or $\G$ is
    a dual polar graph or a Hamming graph,
\item $\G$ has no parallelogram of length $2$ or $3$,
\item $\G$ is $D$-bounded.
\end{enumerate}
\end{prop}

\noindent Liang and Weng \cite{LW97} showed that if $\G$ is $Q$-polynomial and $D\ge 4$ then $\G$ is parallelogram-free
if and only if either (i) $\G$ is bipartite, (ii) $\G$ is a generalized odd graph, or (iii) $\G$ has classical parameters and either $b < -1$ or
$\G$ is a Hamming graph or a dual polar graph.
Weng \cite{W97} showed that if $\G$ has
classical parameters with $b<-1, a_1\neq 0, c_2>1$, and $D \geq 4$, then $\G$
has geometric parameters
(cf.~Section
\ref{sec:generalizations+geometric}).
Building on this, he showed among other results that there are no distance-regular graphs with
classical parameters with $D \ge 4$, $c_2=1$, and $a_2>a_1>1$,
and that under the assumption
$D \ge 4$ and $c_2>1$, the dual polar graphs $^2\A_{2D-1}(-b)$ are the only
graphs with classical parameters with $b=-a_1-1$.
The latter characterizes the
dual polar graphs $^2\A_{2D-1}(\sqrt{q})$ by their intersection arrays for $D \ge 4$.
Weng \cite{We99} also showed the following result.

\begin{theorem}\label{thm:b<-1}
If $\G$ has classical parameters with $b<-1$, $a_1\neq 0$, $c_2>1$, and $D
\geq 4$, then either $\G$ is a dual polar graph $^2\A_{2D-1}(-b)$ or a
Hermitian forms graph $Her(D,(-b)^2)$, or $\alpha=(b-1)/2$, $\beta=-(1+b^D)/2$,
and $-b$ is a power of an odd prime.
\end{theorem}

\noindent
Vanhove \cite{Vanhove2012JAC} showed that
a $((1-b)/2)$-\emph{ovoid} (i.e., a $(D-1)$-design with index $(1-b)/2$) in the
dual polar graph $^2\A_{2D-1}(-b)$ with $b$ odd would induce a distance-regular
graph having classical parameters of the latter case. For $D=2$, such
$((1-b)/2)$-ovoids are better known as \emph{hemisystems} and these were
constructed by Cossidente and Penttila \cite{CP2005JLMS} for every odd prime
power $-b$; see also~\cite{BGR2010BLMS}. No construction of a $((1-b)/2)$-ovoid
is known for $D\ge 3$.

Triangle-free distance regular graphs with classical parameters have been
studied by Pan, Lu, and Weng \cite{PanLuWeng08,PanWeng08,PanWeng09} and Hiraki
\cite{Hiraki09a}. One of the results is that if $\G$ has classical
parameters and $a_1=0, a_2 \neq 0, D\geq 3$ then either (i)
$(b,\alpha,\beta)=(-2,-2,((-2)^{D+1}-1)/3)$ ($c_2=1$), or (ii)
$(b,\alpha,\beta)=(-2,-3,-1-(-2)^D)$ ($c_2=2$), or
(iii)$(b,\alpha,\beta)=(-3,-2,-(1+(-3)^D)/2)$ ($c_2=2$); cf.~\cite{PanWeng09,Hiraki09a}.
Case (i) with $D=3$ is uniquely realized by the Witt graph $M_{23}$ \cite[\S 11.4B]{bcn}, whereas Huang, Pan, and Weng \cite{HPW2009pre} ruled out case (i) with $D\geq 4$.
Case (ii) is uniquely realized by the Hermitian forms graph $\mathit{Her}(D,4)$.



\subsection{Imprimitive graphs with classical parameters and partition graphs}
\label{sec: partition graphs}

It is known (\cite[Prop.~6.3.1]{bcn}) when a distance-regular graph with classical
parameters $(D,b,\alpha,\beta)$ with $D \geq 3$ is imprimitive:
it is bipartite
if and only if $\alpha=0$ and $\beta=1$, whereas it is antipodal if and only if
$b=1$ and $\beta=1+\alpha(D-1)$, in which case it is an antipodal double cover
of its folded graph. This folded graph has diameter $D'$ and intersection
numbers $b_i=(D-i)(1+\alpha(D-1-i))$ and $c_i=i(1+\alpha(i-1))$ for $i<D'$,
$b_{D'}=0$, and $c_{D'}=\gamma D'(1+\alpha(D'-1))$, where $\gamma=1$ if
$D=2D'+1$ and $\gamma=2$ if $D=2D'$.
The distance-regular graphs with such intersection numbers
are called \emph{pseudo partition graphs}.
Bussemaker and Neumaier \cite[Thm.~3.3]{BN1992MC} showed that pseudo
partition graphs with diameter $D' \geq 3$
must have the same intersection
arrays as in one of the three families of partition graphs: the folded cubes
($\alpha=0$), the folded Johnson graphs ($\alpha=1$), and the folded
halved cubes ($\alpha=2$).

The folded cubes are determined by their intersection arrays, except for the
folded $6$-cube, which has two mates (i.e., non-isomorphic distance-regular
graphs with the same intersection array) in the form of (other) incidence
graphs of $2$-$(16,6,2)$ designs (cf.~\cite[Thm.~9.2.7]{bcn}).
The characterization of the other two families of partition graphs is now complete,
due to work by Metsch, Gavrilyuk and Koolen:
\begin{prop}\label{folded Johnson}
{\em \cite{M297,M197,GK2012pre}} The folded Johnson graphs with
diameter at least three are uniquely determined as distance-regular graphs by their
intersection arrays.
\end{prop}

\begin{prop}\label{folded halved cubes}
{\em \cite{M297,Me03,GK2012pre}}
The folded halved cubes with diameter at least three are uniquely determined as distance-regular graphs by their
intersection arrays.
\end{prop}

\noindent
Thus, all pseudo partition graphs with diameter at least three are known.



\subsection{Characterizations of the \texorpdfstring{$Q$-polynomial}{Q-polynomial} property}\label{sec:Qpolcharacterizations}

Bannai and Ito \cite[p.~312]{bi} conjectured that every
\emph{primitive} distance-regular graph with sufficiently large
diameter is $Q$-polynomial. We note that the Doubled Odd graphs are
not $Q$-polynomial yet have arbitrarily large diameter, so that the
`primitivity' condition in the conjecture is necessary. Currently
we know of no (real) progress towards proving the conjecture;
however there are several new characterizations of the
$Q$-polynomial property (since `BCN'
\cite{bcn}). For completeness and because of its importance, we begin with Terwilliger's \emph{balanced set
condition} \cite{Terwilliger1987JCTA,Terwilliger1995DM}; cf.~\cite[\S 2.11, \S
8.3]{bcn}.
For distinct $x,y\in V$ and for $i,j=0,1,\dots,D$, we let
$\chi_{i,j}(x,y)=\sum_{z\in \G_{i,j}(x,y)}\mathbf{e}_z$ denote the
characteristic vector of $\G_{i,j}(x,y)=\G_i(x)\cap \G_j(y)$; cf.~Section \ref{sec:homogeneity4}.

\begin{theorem}{\em (Balanced set condition \cite{Terwilliger1987JCTA,Terwilliger1995DM})}
The primitive idempotent $E$ is $Q$-polynomial if and only if
$u_i\ne 1$  for all $i=1,2,\dots,D$ and
\begin{equation}\label{balanced set condition}
	E\chi_{i,j}(x,y)-E\chi_{j,i}(x,y)=p_{ij}^h\frac{u_i-u_j}{1-u_h}(E\mathbf{e}_x-E\mathbf{e}_y)
\end{equation}
for all $i,j=0,1,\dots,D$, $h=1,2,\dots,D$, and $x,y\in V$ with $d(x,y)=h$.
\end{theorem}

\noindent Terwilliger \cite{Terwilliger1995DM} obtained an inequality for every
$\ell=3,4,\dots,D$ involving only the intersection numbers, $\theta$, and
$(u_i)_{i=0}^D$, by applying Cauchy-Schwarz to
$E\chi_{i,1}(x,y)-E\chi_{1,i}(x,y)$ and $E\mathbf{e}_x-E\mathbf{e}_y$
with $\{i,h\}=\{\ell,\ell-1\}$, and averaging over $x,y\in V$ with
$d(x,y)=h$.
Equality is attained for all $\ell=3,4,\dots,D$ (or just for $\ell=3$) if
and only if $E$ is $Q$-polynomial; cf.~\cite[\S 8.3]{bcn}.
Instead of the four vectors in \eqref{balanced set condition}, we may also
consider the linear dependency of $E\chi_{i,j}(x,y)$, $E\mathbf{e}_x$, and
$E\mathbf{e}_y$. This was worked out in detail by Terwilliger
\cite{Terwilliger1988GC}.\footnote{The term `balanced set' was introduced in \cite{Terwilliger1988GC} in this context, but many authors now refer to \eqref{balanced set condition} as \emph{the} balanced set condition.}
In particular, he applied Cauchy-Schwarz to
$E\chi_{1,1}(x,y)$ and $E\mathbf{e}_x+E\mathbf{e}_y$, and took the average over
each of the sets $\{(x,y):d(x,y)=h\}$ $(h=1,2)$ to obtain an inequality
involving only $a_1,b_1,c_2,u_1,u_2$; in this case, equality is attained if and
only if $E$ is $Q$-polynomial with
$a_0^{\ster}=a_1^{\ster}=\dots=a_{D-1}^{\ster}=0$. The linear dependency among
$E\chi_{1,1}(x,y)$, $E\mathbf{e}_x$, and $E\mathbf{e}_y$ for adjacent $x$ and
$y$ is also relevant to the property of being tight; cf.~Section
\ref{sec:tightDRG}. There is also a `symmetric' version of \eqref{balanced set
condition} due to Terwilliger \cite[Thm.~2.6]{Terwilliger1995EJC}. This lead, in particular, to
the characterization of the Hermitian forms graphs by their intersection
arrays; cf.~Section \ref{sec:recentclassical}. Tonejc \cite{JurTonpre} recently
presented several inequalities by considering the vectors
$E\chi_{i,1}(x,y)+E\chi_{1,i}(x,y)$ and $E\mathbf{e}_x+E\mathbf{e}_y$.

The following result is due to Pascasio \cite{Pascasio2008DM} and may be viewed
as an extension of \cite[Thm.~8.2.1]{bcn} for the bipartite case.

\begin{prop}
\label{Pascasio} $E$ is $Q$-polynomial if and only if all following properties
hold:
\begin{enumerate}[{\em (i)}]
\item there exist $p,r\in\mathbb{C}$ such that $u_{i-1}+u_{i+1}=pu_i+r$
    $(i=1,2,\dots,D-1)$,
\item there exist $\xi,\omega,\eta^{\ster}\in\mathbb{C}$ such that
    $a_i(u_i-u_{i-1})(u_i-u_{i+1})=\xi u_i^2+\omega u_i+\eta^{\ster}$
    $(i=0,1,\dots,D)$, where $u_{-1}$ and $u_{D+1}$ are defined by (i) with
    $i=0$ and $i=D$, respectively,
\item $u_i\ne 1$ $(i=1,2,\dots,D)$.
\end{enumerate}
\end{prop}

\noindent We call $E$ a \emph{tail} \cite{Lang2002EJC} if $E\circ E$ is a
linear combination of $E_0,E$, and at most one other primitive idempotent of
$\AL$. Juri\v{s}i\'{c}, Terwilliger, and \v{Z}itnik \cite{JTZ2010JCTB}
established a characterization similar to Proposition \ref{Pascasio}, where
property (ii) is replaced by $E$ being a tail. We shall discuss tails in detail
in Section \ref{sec:vanishingKrein}.

The following characterization is due to
Kurihara and Nozaki \cite{KN2012JCTA}; cf.~\cite{NT2011LAA}.

\begin{prop}
Let $F$ be a primitive idempotent other than $E$. Then there is a
$Q$-polynomial ordering $(E_i)_{i=0}^D$ such that $E=E_1$ and $F=E_D$ if and
only if $u_0,u_1,\dots,u_D$ are distinct, and for $i=0,1,\dots,D$, the
eigenvalue of $A_i$ for $F$ is
\begin{equation*}
	\frac{(1-u_1)(1-u_2)\cdots(1-u_D)}{(u_i-u_0)\cdots(u_i-u_{i-1})(u_i-u_{i+1})\cdots(u_i-u_D)}.
\end{equation*}
\end{prop}

\noindent
This result
originated in an investigation of the $D$ distances occurring in the spherical
embedding $\{E\mathbf{e}_x\,:\, x\in V\}\subset\mathbb{R}^{m(\theta)}$,
extending a similar observation by Bannai and Bannai \cite{BB2005EJC} for strongly regular graphs.\footnote{See e.g., \cite{CGS1978NAWIM,Munemasa2004EJC,Suda2011JCD} for some results about spherical designs (cf.~\cite{BB2009EJC}) obtained in this way from $Q$-polynomial distance-regular graphs.}
Nozaki \cite{Nozaki2013pre} recently showed that $E$ is $Q$-polynomial provided that $v>\binom{m(\theta)+D-2}{D-1}+\binom{m(\theta)+D-3}{D-2}$ and $u_i\ne 1$ for $i=1,2,\dots,D$.

There are also many results characterizing $Q$-polynomial graphs within certain subclasses of distance-regular graphs,
such as bipartite graphs and tight graphs (cf.~Section \ref{sec:tightDRG}); see, e.g.,
\cite{Terwilliger1995DM,Pascasio2001GC,Lang2004JCTB,TW05,Suzuki2007EJC}. For example, if $\G$ is a thick regular near
polygon with $D\geq 3$, then $\G$ is $Q$-polynomial if and only if $\G$ has classical parameters;
cf.~\cite[Thm.~8.5.1]{bcn}. It should be remarked that De Bruyn and Vanhove \cite{DV2012pre} recently showed that for
$D\geq 4$ there are no $Q$-polynomial thick regular near polygons, apart from the Hamming graphs and dual polar graphs.
See also \cite[Thm.~C]{W97} and Theorem \ref{thm:RNPthick}.


\subsection{Classification results}
\label{Q-classification}

In this section, suppose that $E$ is a $Q$-polynomial idempotent, and let $p,r,r^{\ster}$ be as in \eqref{TTR}.
Note that these scalars depend on $E$.
We note also that, in the notation of Bannai and Ito \cite[\S III.5]{bi}
(cf.~\cite[\S 2]{Talgebra92}), $E$ is classical if and only if the $Q$-polynomial structure satisfies either type I with
$s^*=0$ or one of types IA, IIA, IIC; see \cite[Prop.~6.2]{Tanaka2011EJC} (cf.~\cite[Thm.~8.4.1]{bcn}). It turns out that most
graphs with $p=\pm 2$ already appeared in Sections \ref{sec: the case b=1} and \ref{sec: partition graphs}.


\subsubsection{Case \texorpdfstring{$p\ne\pm2$}{p ne pm 2}}
The $Q$-polynomial structure is type I or type IA in \cite{bi}.
The following result is due to Terwilliger [unpublished].

\begin{prop}
Type IA does not occur.
\end{prop}

\begin{proof}
If the $Q$-polynomial structure is type IA then $E$ is classical and we have
\begin{equation*}
	\theta_i=\theta_0-sb(1-b^i), \quad b_i=-tb^{i+1}(1-b^{i-D}), \quad c_i=b(1-b^i)(s-tb^{i-D-1})
\end{equation*}
for $i=0,1,\dots,D$, where $b,s,t\in\mathbb{C}\backslash\{0\}$ and $p=b+b^{-1}$; cf.~\cite[\S III.5]{bi}, \cite[\S
2]{Talgebra92}.
The corresponding classical parameters are $(D,b,\alpha,\beta)$, where $\alpha=tb^{1-D}(1-b)^2$ and $\beta=tb^{1-D}(1-b)$. In particular, $b$ is an integer distinct from $0,\pm 1$, and thus $s,t\in\mathbb{R}$.
From $\theta_0>\theta_1,\theta_2$, it follows that $b\ge 2$ and $s<0$.
Moreover, because $c_2>0$ we have $sb^D\leq 2sb^{D-1}<2t$.
But then
$\theta_0+\theta_D=2b_0-sb(1-q^D)=b(1-b^{-D})(sb^D-2t)<0$, so that $\theta_D<-\theta_0$, a contradiction.
\end{proof}

\noindent It follows that all graphs having classical parameters with $b\ne 1$ fall into type I with $s^*=0$ (with
respect to the associated $Q$-polynomial ordering).

\subsubsection{Case \texorpdfstring{$p=2$}{p=2}, \texorpdfstring{$r\ne 0$}{r ne 0}, \texorpdfstring{$r^{\ster}\ne 0$}{r* ne 0}}
The $Q$-polynomial structure is type II in \cite{bi}. Terwilliger \cite{Terwilliger1986JCTB} showed that if $D\geq 14$
then either $\G$ is the halved $(2D+1)$-cube, or $\G$ has the same intersection array as a folded Johnson graph or a
folded halved cube. By Propositions \ref{folded Johnson} and \ref{folded halved cubes}, the classification is now
complete for $D\geq 14$.

\subsubsection{Case \texorpdfstring{$p=2$}{p=2}, \texorpdfstring{$r=0$}{r=0}, \texorpdfstring{$r^{\ster}\ne 0$}{r* ne 0}} $E$ is classical and the $Q$-polynomial structure is type IIA in \cite{bi}.
It follows that $\G$ is either a Johnson graph, a halved cube, or the Gosset graph; cf.~Section \ref{sec: the case
b=1}.

\subsubsection{Case \texorpdfstring{$p=2$}{p=2}, \texorpdfstring{$r\ne 0$}{r ne 0}, \texorpdfstring{$r^{\ster}=0$}{r*=0}}
The $Q$-polynomial structure is type IIB in \cite{bi}.
Terwilliger \cite{Terwilliger1988C} showed that $\G$ is either a folded cube or one of the other two non-isomorphic graphs with the intersection array $\{6,5,4;1,2,6\}$ of the folded $6$-cube; cf.~\cite[\S
9.2D]{bcn}.

\subsubsection{Case \texorpdfstring{$p=2$}{p=2}, \texorpdfstring{$r=r^{\ster}=0$}{r=r*=0}}
$E$ is classical and the $Q$-polynomial structure is type IIC in \cite{bi}.
Egawa \cite{E-b1} showed that $\G$ is either a Hamming
graph or a Doob graph; cf.~\cite[\S 9.2B]{bcn}.

\subsubsection{Case \texorpdfstring{$p=-2$}{p=-2}}
The $Q$-polynomial structure is type III in \cite{bi}.
Terwilliger \cite{Terwilliger1987JCTB} showed that $\G$ is either the $D$-cube ($D$
even), the Odd graph $O_{D+1}$, or the folded $(2D+1)$-cube.

\medskip
\noindent Next, we move on to (almost) imprimitive graphs.


\subsubsection{Bipartite graphs}\label{sec:bipartiteDRG}
Suppose $\G$ is bipartite. Then $r^{\ster}=0$ by \cite[Thm.~8.2.1]{bcn}. If $p=\pm 2$ then it follows from the above
results that $\G$ is either the $D$-cube, the folded $2D$-cube, or one of the other two graphs with intersection array
$\{6,5,4;1,2,6\}$. Caughman \cite{Caughman2004GC} showed that if $p\ne\pm 2$ and $D\geq 12$ then $\G$ has classical
parameters $(D,b,0,1)$ where $b$ is an integer at least $2$.
These parameters are realized by the dual polar graphs $\mathcal{D}_D(b)$ and the Hemmeter graphs.


\subsubsection{Antipodal graphs}\label{sec:antipodalDRG}
Curtin \cite{Curtin1998DM} showed that bipartite $Q$-polynomial antipodal (double) covers are precisely the bipartite
$2$-homogeneous distance-regular graphs, and the latter graphs were classified by Nomura \cite{Nomura1995JCTB};
cf.~Section \ref{sec:homogeneity}. These are the $D$-cube, the regular complete bipartite graphs minus a perfect
matching,
the Hadamard graphs,
and the graphs with intersection arrays satisfying
\begin{equation*}
	(c_1,c_2,\dots,c_5)=(1,\mu,k-\mu,k-1,k), \quad b_i=c_{5-i} \ (i=0,1,\dots,4),
\end{equation*}
where $k=\gamma(\gamma^2+3\gamma+1)$, $\mu=\gamma(\gamma+1)$, and $\gamma\geq 2$ is an integer. The last case is
uniquely realized for $\gamma=2$ by the double cover of the Higman-Sims graph.

Dickie and Terwilliger \cite{DT1996EJC} gave a classification of non-bipartite $Q$-polynomial antipodal
distance-regular graphs as follows: the Johnson graph $J(2D,D)$, the halved $2D$-cube, the
non-bipartite Taylor graphs, and the graphs satisfying
\begin{gather}
	(c_1,c_2,c_3,c_4)=(1,p\eta,(p^2-1)(2\eta-p+1), p(2\eta+2\eta p-p^2)), \label{Q-AT4} \\
	b_i=c_{4-i} \ (i=0,1,2,3), \notag
\end{gather}
where $p\geq 3$, $\eta\geq 3p/4$ are integers and $\eta$ divides $p^2(p^2-1)/2$.
An example of the last case is the Meixner double cover ($p=4,\eta=6$); cf.~Section \ref{sec: Soicher and Meixner}. The
array \eqref{Q-AT4} with $p,\eta$ odd has been ruled out by Juri\v{s}i\'{c} and Koolen \cite[Cor.~3.2]{JuKo00EuJC}.


\subsubsection{Almost bipartite graphs}\label{sec:Qalmostbipartite}
The $Q$-polynomial generalized odd graphs have been classified by Lang and Terwilliger \cite{LT2007EJC}:
the folded $(2D+1)$-cube, the Odd graph $O_{D+1}$, and the graphs with $D=3$ satisfying
\begin{equation*}
	k=1+(p^2-1)(p(p+2)-(p+1)c_2), \quad c_3=-(p+1)(p^2+p-1-(p+1)c_2),
\end{equation*}
where $p<-2$ is an integer. No example is known for the last case.
We recall that the distance-$2$ graph $\G_2$ is again distance-regular, as it is the halved graph of
the bipartite double of $\G$; cf.~Section \ref{sec:2Porder}.


\subsubsection{Almost \texorpdfstring{$Q$-bipartite}{Q-bipartite} graphs}
Suppose $D\geq 4$ and $\G$ is almost $Q$-bipartite, i.e., $a_i^{\ster}=0$ for $i<D$ and $a_D^{\ster}>0$. Dickie
\cite{Dickie1995D} showed that $\G$ is either the halved $(2D+1)$-cube, the folded $(2D+1)$-cube, or a dual polar graph
$^2\mathcal{A}_{2D-1}(\sqrt{q})$. We note that the `$Q$-bipartite double' of $\G$ is a cometric association
scheme, and that $E_2$ is again a $Q$-polynomial idempotent; cf.~Sections \ref{sec:Qmultipleordering} and
\ref{sec:cometricschemes}.


\subsection{The Terwilliger algebras of \texorpdfstring{$Q$-polynomial}{Q-polynomial} distance-regular graphs}
\label{sec:thinness-Q}

Below we collect `handy' sufficient conditions for $\G$ being thin when it is $Q$-polynomial.

\begin{prop}{\em \cite[\S 5]{Talgebra92}}\label{thin criterion}
Suppose $\G$ is $Q$-polynomial with respect to the ordering $(E_i)_{i=0}^D$.
Then the following properties hold.
\begin{enumerate}[{\em (i)}]
\item $\G$ is thin with respect to $x\in V$ if for $i=1,2,\dots,D$ and for every $y,z\in\G_i(x)$, there is an
    automorphism $\pi$ of $\G$ such that $\pi(x)=x$, $\pi(y)=z$, and $\pi(z)=y$,
\item $\G$ is thin if $a_2=a_3=\dots=a_{D-1}=0$,
\item $\G$ is thin if $a_2^{\ster}=a_3^{\ster}=\dots=a_{D-1}^{\ster}=0$.
\end{enumerate}
\end{prop}

\noindent In particular, a $Q$-polynomial distance-regular graph is thin provided that it is bipartite (=$Q$-antipodal), almost bipartite, antipodal (=$Q$-bipartite), or almost $Q$-bipartite. It also follows that
many of the known graphs with classical parameters as well as partition graphs (cf.~Section \ref{sec: partition
graphs}) are thin; see \cite[Ex.~6.1]{Talgebra92} for details. The following graphs are known to be \emph{non}-thin:
Doob graphs, (bilinear, alternating, Hermitian, quadratic) forms graphs, and the twisted Grassmann graphs. The
irreducible $\TT$-modules of the Doob graphs were determined by Tanabe \cite{Tanabe1997JAC}. Concerning the twisted
Grassmann graph (cf.~Section \ref{twistedsection}), Bang, Fujisaki, and Koolen \cite{BFK09} showed that it is thin with
respect to any base vertex $x$ which is an $(e-1)$-dimensional subspace of the fixed hyperplane $H$, by verifying a
different combinatorial criterion for thinness \cite[Thm.~5.1(v)]{Talgebra92}. However, they also showed that if $x$ is
not contained in $H$ then the twisted Grassmann graph is not $1$-thin with respect to $x$.

The irreducible $\TT$-modules of bipartite (resp.~almost bipartite) $Q$-polynomial distance-regular graphs were
described by Caughman \cite{Caughman1999DM} (resp.~Caughman, MacLean, and Terwilliger~\cite{CMT2005DM}). For
these graphs, it turns out that the intersection array completely determines the structure of $\TT$. In particular,
explicit formulas for the multiplicities of the irreducible $\TT$-modules in $\mathbb{C}^v$ with small endpoints were
successfully used in the classification of these graphs; cf.~Section \ref{Q-classification}. Curtin and Nomura
\cite{CN2000JAC} and Curtin \cite{Curtin2001JCTB} studied the Terwilliger algebra of bipartite $Q$-polynomial antipodal
(double) covers which are not the $D$-cube; in this case, it follows that $\TT$ is a homomorphic image of the quantum
enveloping algebra $U_q(\mathfrak{sl}_2)$; cf.~Section \ref{sec: TD systems}.

In general, if $\G$ is $Q$-polynomial then the structure of irreducible $\TT$-modules with endpoint $1$ is determined
by the intersection array and the spectrum of the local graph $\Upsilon(x)$ with respect to the base vertex $x$;
cf.~\cite[Lecture 35]{Terwilliger1993N}. To be more precise, suppose for the moment that $\G$ is $Q$-polynomial, and
let $W$ be an irreducible $\TT$-module with endpoint $1$. Then $\dim E_1^{\ster}W=1$, so that $E_1^{\ster}W$ is an
eigenspace for $E_1^{\ster}AE_1^{\ster}$; let $\eta$ denote the corresponding eigenvalue. Then the isomorphism class of
$W$ is determined by $\eta$. Moreover, $W$ is thin if and only if $\eta$ is a root of a polynomial $T$ of degree $4$,
which we call the \emph{Terwilliger polynomial} of $\G$;
if $\G$ has classical parameters $(D,b,\alpha,\beta)$ then its four roots are $-1,-b-1,\beta-\alpha-1$, and $\alpha b\gauss{D-1}{1}-1$.
See also \cite[Lemma 4.7]{GK2012pre} and \cite[Cor.~4.12(5)]{Talgebra92}.
If $W$ is non-thin then it follows that\footnote{In fact, Terwilliger \cite[Lectures 34--37]{Terwilliger1993N} stated this result as a conjecture, and showed that $\dim E_i^{\ster}W\leq 2$ for $i=2,3,\dots,D-1$, $\dim E_D^{\ster}W\leq 1$, and that $W$ is thin if $\dim E_2^{\ster}W=1$. Now, that $W$ has diameter $D-1$ follows from a result of Go and Terwilliger \cite[Thm.~9.8]{GT2002EJC}, and the values of the $\dim E_i^{\ster}W$ follow from their symmetric and unimodal properties proved by Ito, Tanabe, and Terwilliger \cite{ITT2001P} in the more general context of tridiagonal systems; cf.~Section \ref{sec: TD systems}.}
$W$ has diameter $D-1$ and that $\dim E_1^{\ster}W=\dim E_D^{\ster}W=1$, $\dim E_2^{\ster}W=\dots=\dim E_{D-1}^{\ster}W=2$.
Hobart and Ito \cite{HI1998JAC} studied in detail the structure of such a non-thin irreducible $\TT$-module with
endpoint $1$. Miklavi\v{c} \cite{Miklavic2004EJC,Miklavic2009EJC} showed that $\G$ is $1$-homogeneous if it is
$Q$-polynomial with $a_1=0$, and described the unique irreducible $\TT$-module with endpoint $1$ (with $\eta=0$) when
$a_2\ne 0$, which turns out to be non-thin. Miklavi\v{c} \cite{Mik09} also described the irreducible $\TT$-modules with
endpoint $1$ when $\G$ has classical parameters with $b<-1$, $a_1\ne 0$, and is not a near polygon; there are exactly
two isomorphism classes, and the first one is thin with $\eta=-1$ and the second one is non-thin with $\eta=a_1$.

Suppose again that $\G$ is $Q$-polynomial, and let $\eta$ be a local eigenvalue of $\G$ (with respect to the base
vertex $x$), i.e., an eigenvalue of $\Upsilon(x)$. We call $\eta$ \emph{non-degenerate} if it has an eigenvector
orthogonal to the all-ones vector, and \emph{degenerate} otherwise. We note that $a_1$ is the only possible degenerate local eigenvalue and that it is
non-degenerate precisely when $\Upsilon(x)$ is disconnected. The Terwilliger polynomial $T$ mentioned above depends
only on the intersection array of $\G$ and the $Q$-polynomial ordering, and has the property that $T(\eta) \geq 0$
for every non-degenerate local eigenvalue $\eta$ for every base vertex $x$.
We note that if $\G$ has two $Q$-polynomial orderings then $T$ may be different for the different ordering.
Using the polynomial $T$, Gavrilyuk and Koolen \cite{GK2012pre} recently showed the uniqueness of the folded halved
$2m$-cube for $m \geq 6$; cf.~Section \ref{sec: partition graphs}. With the same approach we can also show the
uniqueness of the folded Johnson graphs. For the Grassmann graphs $J_q(2D, D)$ $(D \geq 3)$, Gavrilyuk and Koolen also
obtained partial results.
See also \cite[\S 4.3]{TTW16+} for more discussions on the Terwilliger polynomial.

See Section \ref{sec:thinness} for more results on the irreducible $\TT$-modules with endpoint $1$ of general
distance-regular graphs.


\subsection{Further results on \texorpdfstring{$Q$-polynomial}{Q-polynomial} distance-regular graphs}
\label{sec: further results}

In this section, we always assume that $\G$ is $Q$-polynomial.

\subsubsection{Antipodal covers}
Van Bon and Brouwer \cite{BB1988P} determined the distance-regular antipodal covers of the classical families of
distance-regular graphs; cf.~\cite[\S 6.12]{bcn}. Suppose $E$ is $Q$-polynomial, and recall (cf.~\eqref{TTR}) that
there exist $p,r\in\mathbb{C}$ such that $u_{i-1}+u_{i+1}=pu_i+r$ for $i=1,2,\dots,D-1$. Terwilliger
\cite{Terwilliger1993EJC} showed that if $\G$ has an antipodal cover of diameter $\tilde{D}\geq 7$, then this
three-term recurrence extends to $i=1,2,\dots,\tilde{D}-1$, where we formally define $u_i=u_{\tilde{D}-i}$
($i=D+1,D+2,\dots,\tilde{D}$). This parametric condition provides simple proofs of some of the (non-existence) results
in \cite{BB1988P}, and may be applied to the twisted Grassmann graphs as well; cf.~\cite{FKT07}. Caughman
\cite{Caughman1999JCTB} used the condition to show that if $\G$ is bipartite with $D\geq 4$ and has an antipodal cover
then $\G$ is the folded $2D$-cube; cf.~\cite[Cor.~12.3]{Lang2004JCTB}.


\subsubsection{Distance-regular graphs with multiple \texorpdfstring{$Q$-polynomial}{Q-polynomial} orderings}\label{sec:Qmultipleordering}
An association scheme can have at most two $P$-polynomial orderings, except for those coming from the polygons; cf.~\cite[\S 4.2D]{bcn}.
Bannai and Ito \cite[pp.~354--360]{bi} showed that if $k\geq 3$ and $D\geq 34$ then $\G$ has at most two $Q$-polynomial idempotents and moreover all eigenvalues are integral.
Brouwer, Cohen, and Neumaier \cite[p.~247]{bcn} conjectured
that the assumption $D\geq 34$ can be replaced by $D\ne 4$. Dickie
\cite[pp.~69--70]{Dickie1995D} established the result under the assumption $D\geq 5$.
Indeed, he showed that if $k\geq 3$ and $D\geq 5$ then $\G$ has more than one $Q$-polynomial idempotent if and only if $\G$ is either the $D$-cube ($D$ even), the halved $(2D+1)$-cube, the folded $(2D+1)$-cube, or a dual polar graph
$^2\!\mathcal{A}_{2D-1}(\sqrt{q})$, and these graphs have precisely two $Q$-polynomial idempotents but no non-integral eigenvalues.
(Note that if $\G$ has non-integral eigenvalues and $E$ is $Q$-polynomial then $E^{\sigma}$ is again $Q$-polynomial for any $\mathbb{Q}$-automorphism $\sigma$ of the splitting field over $\mathbb{Q}$.)
Building on work by
Dickie \cite{Dickie1995D}, Suzuki \cite{Suzuki1998JACb} showed that every association scheme has at most two
$Q$-polynomial idempotents, again except for those coming from the polygons; cf.~Section \ref{sec:cometricschemes}.
For $D\in\{2,3,4\}$, the known $Q$-polynomial distance-regular graphs with $k\geq 3$ and with non-integral eigenvalues belong to the following four families:
the conference graphs $(D=2)$, the incidence graphs of symmetric designs ($D=3$),
the Taylor graphs ($D=3$), and the Hadamard graphs ($D=4$).\footnote{It seems that the above conjecture by Brouwer et al.~was not properly stated, because we already have counterexamples with $D\in\{2,3\}$. We note that the graphs in the last three families are imprimitive.}
Note that the graphs in these families always have two $Q$-polynomial idempotents.
The other candidate intersection arrays $\{\mu(2\mu+1),(\mu-1)(2\mu+1),\mu^2,\mu;1,\mu,\mu(\mu-1),\mu(2\mu+1)\}$ ($\mu\geq 2$) of primitive $Q$-polynomial distance-regular graphs with non-integral eigenvalues given by Brouwer et al.~\cite[pp.~247--248]{bcn} were ruled out by Godsil and Koolen \cite{GoKo95}; cf.~Section \ref{sec:tablesD4prnon}.
Ma and Koolen \cite{KM2014pre} recently classified the distance-regular graphs with $k\geq 3$, $D=4$, and with two $Q$-polynomial idempotents; these are the $4$-cube, the halved $9$-cube, the folded $9$-cube, the dual polar graphs $^2\!\mathcal{A}_7(\sqrt{q})$, and the Hadamard graphs.


\subsubsection{Bounds for the girth}\label{sec:girth}
Brouwer, Cohen, and Neumaier \cite[p.~248]{bcn} conjectured that $\G$ has girth at most $6$, with equality only for the Odd graph $O_{D+1}$, and showed that the numerical girth $g$ of $\G$ is at most $7$.
Lewis \cite{Lewis2000DM} showed $c_3\geq 2$,
proving $g\leq 6$.
We note that if $\G$ has girth $6$, i.e., $a_1=a_2=0$ and $c_2=1$, then it follows from Proposition \ref{Pascasio} (or
\cite[Thm.~6.3]{Miklavic2004EJC}) that $a_1=a_2=\dots=a_{D-1}=0$, so that $\G$ is bipartite or almost
bipartite.
Miklavi\v{c} \cite{Miklavic2007DM} showed that if $\G$ is bipartite and $D=4$ then $c_2\geq 2$, i.e., $g=4$.


\subsubsection{The \texorpdfstring{Erd\H{o}s-Ko-Rado}{Erdos-Ko-Rado} theorem}\label{sec:EKR}
At the end of each of Sections 9.1-9.4 and 9.5A in \cite{bcn} there is a remark about the Erd\H{o}s-Ko-Rado theorem
for the graph in question. See
\cite{Tanaka2006JCTA,PSV2011JCTA,Tanaka2010pre,IM2013DCC,Tanaka2012pre,GM2016B}
for recent results on this topic.


\subsubsection{Unimodality of the multiplicities}\label{sec: unimodality of mi}
We recall from Proposition \ref{prop:unimodal} (iv) that the $k_i$ are unimodal. Concerning the multiplicities $m_i$,
Pascasio \cite{Pascasio2002EJC} showed that if $\G$ is $Q$-polynomial with respect to the ordering $(E_i)_{i=0}^D$ then
$m_{i-1}\le m_i\le m_{D-i}$ for $i=1,2,\dots,\lfloor D/2\rfloor$. This
result was originally conjectured by Dennis Stanton in 1993, and is a simple application of the theory of
tridiagonal systems; cf.~Section \ref{sec: TD systems}. We note that Bannai and Ito \cite[p.~205]{bi} earlier
conjectured that the multiplicities of a cometric association scheme satisfy the unimodal property.



\subsubsection{Posets associated with \texorpdfstring{$Q$-polynomial}{Q-polynomial} distance-regular graphs}
\label{sec: posets}

There are several classes of finite ranked posets that are closely related to $Q$-polynomial distance-regular graphs:
\emph{regular semilattices} \cite{Delsarte1976JCTA,Stanton1985JCTA}, \emph{uniform posets} \cite{Terwilliger1990P},
\emph{quantum matroids} \cite{Terwilliger1996P}. (For definitions, see the references given.)
Many of the known families of $Q$-polynomial distance-regular graphs arise as the top fibers of these posets, where two
vertices are adjacent if and only if they cover a common element.

Concerning quantum matroids, Terwilliger \cite[Thm.~38.2]{Terwilliger1996P} showed that if a quantum matroid is
`non-trivial'
 and `regular', then the graph on the top fiber with the above adjacency is distance-regular.
Moreover, in this case, the graph has classical parameters if its diameter is equal to the rank of the quantum matroid.
The culmination of the study of quantum matroids is the classification (\cite[Thm.~39.6]{Terwilliger1996P}) of
non-trivial regular quantum matroids with rank at least four: they are precisely those posets naturally associated with
Johnson, Hamming, Grassmann, bilinear forms, and dual polar graphs. We may use this classification as follows.

Fix a $Q$-polynomial ordering $(E_i)_{i=0}^D$ of $\G$. Let $Y$ be a non-empty subset of $V$ and let $\chi$ be its
characteristic vector. Brouwer, Godsil, Koolen, and Martin~\cite{BGKM03} defined the \emph{width} and \emph{dual width}
of $Y$ by $w=\max\{i:\chi^{\mathsf{T}}A_i\chi\ne 0\}$ and $w^{\ster}=\max\{i:\chi^{\textsf{T}}E_i\chi\ne 0\}$,
respectively. They showed among other results that $w+w^{\ster}\ge D$, and we call $Y$ a \emph{descendent}
(cf.~\cite{Tanaka2011EJC}) of $\G$ if equality holds. It follows that every descendent is completely regular, and that the
induced subgraph is a $Q$-polynomial distance-regular graph if it is connected;
cf.~\cite[Thm.~1--3]{BGKM03}.\footnote{The results in \cite{BGKM03} are in contrast with Delsarte theory \cite{del}
based on the minimum distance and (maximum) strength of a subset. We may remark that Suda \cite{Suda2012JCTA} recently
developed a theory which unifies and `interpolates' some of the theorems in \cite{del} and \cite{BGKM03} to a certain
extent.} We say that a set $\mathscr{D}$ of descendents of $\G$ \emph{satisfies} $\mathrm{(UD)}_i$ if each two vertices $x,y\in
V$ at distance $i$ are contained in a unique descendent in $\mathscr{D}$ with width $i$.

\begin{prop}{\em \cite{Tanaka2011EJC}}
Let $\mathscr{D}$ be a set of descendents of $\G$. Suppose that the following properties hold.
\begin{enumerate}[{\em (i)}]
\item $\G$ has classical parameters,
\item $\mathscr{D}$ satisfies $\mathrm{(UD)}_i$ for all $i$,
\item $Y_1\cap Y_2\in\mathscr{D}\cup\{\emptyset\}$ for all $Y_1,Y_2\in\mathscr{D}$.
\end{enumerate}
Then $\mathscr{D}$, together with the partial order defined by reverse inclusion, forms a non-trivial regular quantum
matroid. In particular, if $D\geqslant 4$ then $\G$ is either a Johnson, Hamming, Grassmann, bilinear forms, or dual
polar graph.
\end{prop}

\noindent It was also shown that if $\mathscr{D}$ is the set of \emph{all} descendents of $\G$ then condition (iii) in
the above proposition is implied by the other two. See \cite{BGKM03,Tanaka2006JCTA,HS2007EJC,Tanaka2011EJC} for more information
on descendents.

Unlike regular semilattices and quantum matroids, uniform posets are not assumed to be semilattices, but give rise to
at least 13 infinite families of $Q$-polynomial distance-regular graphs with unbounded diameter, rather than just five
as above; cf.~\cite[\S 4]{Terwilliger1990P}.
Suppose $\G$ is ($Q$-polynomial and) bipartite, and fix $x\in V$. Then we may view $\G$ as the Hasse diagram
of a ranked poset with $D+1$ fibers $\G_i(x)$ $(i=0,1,\dots,D)$. Miklavi\v{c} and Terwilliger \cite{MT2011pre} recently
showed that this poset is uniform.\footnote{See \cite{MT2011pre} for the precise statement of the result, noting that
the hypercube $H(D,2)$ with $D$ even has two $Q$-polynomial structures. They also introduced the concept of
\emph{strongly uniform} and investigated when the poset $\G$ has that property.} Caughman \cite{Caughman2003EJC} showed
that the graph on the top fiber $\G_D(x)$ defined in the previous manner (which is in this case the induced subgraph of
the distance-$2$ graph of $\G$) is distance-regular and $Q$-polynomial.
See \cite{TW2013AMC} and the references therein for more results on uniform posets.

The poset $\mathscr{S}$ consisting of all strongly closed subgraphs of $\G$ with partial order defined by reverse
inclusion plays an important role in the study of distance-regular graphs having classical parameters with $b<-1$.
Suppose $\G$ has geometric parameters $(D,b,\alpha)$ (cf.~Section \ref{sec:generalizations+geometric}) with $D\geq 4$ and is
$D$-bounded in the sense of Weng \cite{W97,We99}, i.e., every $\Delta\in\mathscr{S}$ is assumed to be regular. Then
$b<-1$ by \cite[Lemma~5.5]{We99}. (Conversely, if $\G$ has classical parameters with $b<-1,D\geq 4,a_1\ne 0,c_2>1$ then
$\G$ is $D$-bounded and has geometric parameters; cf.~\cite[Thm.~5.7,~5.8]{We99}.) In this case, Weng \cite{W97} showed
that $\mathscr{S}$ is a ranked (meet) semilattice and every interval is a modular atomic lattice which is isomorphic to
a projective geometry over $GF(b^2)$.


\subsection{Tridiagonal systems}
\label{sec: TD systems}

Let $W$ be a finite dimensional vector space over $\mathbb{C}$.
Let $\mathfrak{a}\in\mathrm{End}_{\mathbb{C}}(W)$ be diagonalizable, and let $(\theta_i)_{i=0}^{\delta}$ be an ordering of the distinct eigenvalues of $\mathfrak{a}$.
Then there is a sequence of elements $(\mathfrak{e}_i)_{i=0}^{\delta}$ in $\mathrm{End}_{\mathbb{C}}(W)$ such that
(i) $\mathfrak{ae}_i=\theta_i\mathfrak{e}_i$;
(ii) $\mathfrak{e}_i\mathfrak{e}_j=\delta_{ij}\mathfrak{e}_i$;
(iii) $\sum_{i=0}^{\delta}\mathfrak{e}_i=\mathfrak{1}$, where $\mathfrak{1}$ is the identity element in $\mathrm{End}_{\mathbb{C}}(W)$.
(Specifically, $\mathfrak{e}_i=\prod_{j\ne i}\frac{\mathfrak{a}-\theta_j\mathfrak{1}}{\theta_i-\theta_j}$ ($i=0,1,\dots,\delta$).)
We call $\mathfrak{e}_i$ the \emph{primitive idempotent} of $\mathfrak{a}$ associated with $\theta_i$ ($i=0,1,\dots,\delta$).
Let $\mathfrak{a}^{\ster}$ be another diagonalizable element in $\mathrm{End}_{\mathbb{C}}(W)$.
Let $(\theta_i^{\ster})_{i=0}^{\delta^{\ster}}$ be an ordering of the distinct eigenvalues of $\mathfrak{a}^{\ster}$ and let $(\mathfrak{e}_i^{\ster})_{i=0}^{\delta^{\ster}}$ be the corresponding sequence of the primitive idempotents.
The sequence
$\Phi=(\mathfrak{a}; \mathfrak{a}^{\ster}; (\mathfrak{e}_i)_{i=0}^{\delta};
(\mathfrak{e}_i^{\ster})_{i=0}^{\delta^{\ster}})$ is a \emph{tridiagonal system} (or \emph{TD system}) if
\begin{align*}
	\mathfrak{e}_i^{\ster}\mathfrak{a}\mathfrak{e}_j^{\ster}&=0 \quad \text{if } |i-j|>1 \quad (i,j=0,1,\dots,\delta^{\ster}), \\
	\mathfrak{e}_i\mathfrak{a}^{\ster}\mathfrak{e}_j&=0 \quad \text{if } |i-j|>1 \quad (i,j=0,1,\dots,\delta),
%	\mathfrak{e}_i^{\ster}\mathfrak{a}\mathfrak{e}_j^{\ster}&=\begin{cases} 0 & \text{if } |i-j|>1 \\ \ne 0 & \text{if } |i-j|=1 \end{cases} \quad (i,j=0,1,\dots,\delta^{\ster}), \\
%	\mathfrak{e}_i\mathfrak{a}^{\ster}\mathfrak{e}_j&=\begin{cases} 0 & \text{if } |i-j|>1 \\ \ne 0 & \text{if } |i-j|=1 \end{cases} \quad (i,j=0,1,\dots,\delta),
\end{align*}
and $W$ is irreducible as a $\mathbb{C}[\mathfrak{a},\mathfrak{a}^{\ster}]$-module.
This definition is due to Ito, Tanabe, and Terwilliger \cite{ITT2001P}.\footnote{TD systems can be defined on vector spaces over arbitrary fields, and many of the results are valid over wider classes of fields. However, for simplicity and in view of the connections to the theory of $Q$-polynomial distance-regular graphs, we shall only discuss TD systems over $\mathbb{C}$.}
Note that if $\G$ is a $Q$-polynomial distance-regular graph then it follows from \eqref{relations} that every irreducible $\TT$-module naturally has the structure of a TD system.

Suppose that $\Phi$ is a TD system.
Ito et al.~\cite{ITT2001P} showed $\delta=\delta^{\ster}$.
Define $U_i=(\sum_{h=0}^i\mathfrak{e}_h^{\ster}W)\cap(\sum_{\ell=i}^{\delta}\mathfrak{e}_{\ell}W)$ ($i=0,1,\dots,\delta$).
Note that $U_0=\mathfrak{e}_0^{\ster}W$, and that $(\mathfrak{a}-\theta_i\mathfrak{1})U_i\subseteq U_{i+1}$, $(\mathfrak{a}^{\ster}-\theta_i^{\ster}\mathfrak{1})U_i\subseteq U_{i-1}$ ($i=0,1,\dots,\delta$), where $U_{-1}=U_{\delta+1}=0$.
They showed $W=\bigoplus_{i=0}^{\delta}U_i$.
It also turns out that $\dim\mathfrak{e}_iW=\dim\mathfrak{e}_i^{\ster}W=\dim U_i$ $(i=0,1,\dots,\delta)$.
The sum $W=\bigoplus_{i=0}^{\delta}U_i$ is called the \emph{split decomposition}
and plays a crucial role in the theory of TD systems.
Let $\rho_i=\dim\mathfrak{e}_iW$ $(i=0,1,\dots,\delta)$ and call the sequence $(\rho_i)_{i=0}^{\delta}$ the \emph{shape} of $\Phi$.
They showed that the shape is symmetric and unimodal: $\rho_i=\rho_{\delta-i}$
($i=0,1,\dots,\delta$) and $\rho_{i-1}\le \rho_i$ ($i=1,2,\dots,\lfloor \delta/2\rfloor$).
A TD system with $\rho_0=\dots=\rho_{\delta}=1$ is called a \emph{Leonard system} \cite{Terwilliger2001LAA}.
Leonard systems provide a linear algebraic framework for Leonard's theorem and have been extensively studied; see \cite{Terwilliger2006N} and the references therein.
Note that if the TD system $\Phi$ is afforded on an irreducible $\TT$-module of a $Q$-polynomial distance-regular graph, then the $\TT$-module $W$ is thin if and only if $\Phi$ is a Leonard system.
See \cite{Cerzo2010LAA} for a detailed description of thin irreducible $\TT$-modules motivated by the theory of Leonard systems.

Ito et al.~\cite{ITT2001P} showed that there exist scalars $p,\gamma,\gamma^{\ster},\varrho,\varrho^{\ster}\in\mathbb{C}$ such that
\begin{align}
	 0&=[\mathfrak{a},\mathfrak{a}^2\mathfrak{a}^{\ster}-p\mathfrak{a}\mathfrak{a}^{\ster}\mathfrak{a}+\mathfrak{a}^{\ster}\mathfrak{a}^2-\gamma(\mathfrak{a}\mathfrak{a}^{\ster}+\mathfrak{a}^{\ster}\mathfrak{a})-\varrho\mathfrak{a}^{\ster}], \label{TD relation} \\
	 0&=[\mathfrak{a}^{\ster},\mathfrak{a}^{\ster 2}\mathfrak{a}-p\mathfrak{a}^{\ster}\mathfrak{a}\mathfrak{a}^{\ster}+\mathfrak{a}\mathfrak{a}^{\ster 2}-\gamma^{\ster}(\mathfrak{a}^{\ster}\mathfrak{a}+\mathfrak{a}\mathfrak{a}^{\ster})-\varrho^{\ster}\mathfrak{a}], \label{TD relation*}
\end{align}
where $[\mathfrak{b},\mathfrak{c}]:=\mathfrak{bc}-\mathfrak{cb}$, and (cf.~\eqref{TTR})
\begin{equation}\label{AW sequence}
	\frac{\theta_{i-2}-\theta_{i+1}}{\theta_{i-1}-\theta_i}=\frac{\theta_{i-2}^{\ster}-\theta_{i+1}^{\ster}}{\theta_{i-1}^{\ster}-\theta_i^{\ster}}=p+1 \quad (i=2,3,\dots,\delta-1).
\end{equation}
The relations \eqref{TD relation} and \eqref{TD relation*} generalize the $q$-Serre relations (which are among the defining relations of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2))$ and the Dolan-Grady relations (which are the defining relations of the Onsager algebra); cf.~\cite{Terwilliger2001P}.
It is conjectured (\cite[Conj.~13.7]{ITT2001P}) that there exist positive integers $\delta_1,\delta_2,\dots,\delta_n$ such that $\sum_{i=0}^{\delta}\rho_it^i=\prod_{j=1}^n(1+t+\dots+t^{\delta_j})$, where $t$ is an indeterminate.
This conjecture in fact suggests that $\Phi$ would be regarded as a `tensor product' of Leonard systems.
Let $q$ be a nonzero scalar in $\mathbb{C}$ such that $p=q^2+q^{-2}$.
Using the representation theory of $U_q(\widehat{\mathfrak{sl}}_2)$ (cf.~\cite{CP1991CMP}), Ito and Terwilliger \cite{IT2007JAA,IT2010KJM} indeed constructed \emph{all} TD systems (up to isomorphism\footnote{A TD system $\Phi'=(\mathfrak{a}'; \mathfrak{a}^{\ster \prime}; (\mathfrak{e}_i')_{i=0}^{\delta}; (\mathfrak{e}_i^{\ster \prime})_{i=0}^{\delta})$ on a vector space $W'$ is \emph{isomorphic} to $\Phi$ if there is an isomorphism of vector spaces $\sigma:W\rightarrow W'$ such that $\sigma\mathfrak{a}=\mathfrak{a}'\sigma$, $\sigma\mathfrak{a}^{\ster}=\mathfrak{a}^{\ster \prime}\sigma$, and $\sigma\mathfrak{e}_i=\mathfrak{e}_i'\sigma$, $\sigma\mathfrak{e}_i^{\ster}=\mathfrak{e}_i^{\ster \prime}\sigma$ for $i=0,1,\dots,\delta$.}) explicitly as tensor products of Leonard systems (i.e., \emph{evaluation modules}), under the assumption that $q$ is not a root of unity.
We remark that in this case the split decomposition corresponds to the weight space decomposition.
See also \cite{IT2004JPAA,IT2007CA,IT2009JCISS,IT2009JA,Funk-Neubauer2009LAA,HI2012pre}.
Ito [private communication] pointed out that the proofs of most of the results in \cite{IT2010KJM} work under the weaker assumption $q^2\ne\pm 1$, i.e., $p\ne\pm 2$.
It seems that the above conjecture is still open for general TD systems, but Nomura and Terwilliger \cite{NT2008LAAd,NT2010LAAa} showed among other results that $\rho_0=1$, and more generally, $\rho_i\leq \binom{\delta}{i}$ ($i=0,1,\dots,\delta$), a result which would follow directly from the conjecture.
See, e.g., \cite{Hartwig2007LAA,IT2007LAA,IS2014LAA} for some results on TD systems with $p=2$.

Observe now that the $1$-dimensional subspace $\mathfrak{e}_0^{\ster}W$ is invariant under
\begin{equation*}
	(\mathfrak{a}^{\ster}-\theta_1^{\ster}\mathfrak{1})(\mathfrak{a}^{\ster}-\theta_2^{\ster}\mathfrak{1})\dots(\mathfrak{a}^{\ster}-\theta_i^{\ster}\mathfrak{1})(\mathfrak{a}-\theta_{i-1}\mathfrak{1})\dots(\mathfrak{a}-\theta_1\mathfrak{1})(\mathfrak{a}-\theta_0\mathfrak{1})
\end{equation*}
for $i=0,1,\dots,\delta$, and let $\zeta_i$ be the corresponding eigenvalue ($i=0,1,\dots,\delta$).
The sequence $((\theta_i)_{i=0}^{\delta};(\theta_i^{\ster})_{i=0}^{\delta};(\zeta_i)_{i=0}^{\delta})$ is called the \emph{parameter array} of $\Phi$.
Nomura and Terwilliger \cite{NT2008LAAd} showed that the parameter array is a complete invariant for a TD system.
Ito, Nomura, and Terwilliger \cite{INT2011LAA} established the following theorem:

\begin{theorem}{\em \cite[Thm. 3.1]{INT2011LAA}}\label{classification of sharp TD systems}
Let $\pi=((\theta_i)_{i=0}^{\delta};(\theta_i^{\ster})_{i=0}^{\delta};(\zeta_i)_{i=0}^{\delta})$ be a sequence of scalars in $\mathbb{C}$ such that $\theta_i\ne\theta_j$, $\theta_i^{\ster}\ne\theta_j^{\ster}$ if $i\ne j$ $(i,j=0,1,\dots,\delta)$, and suppose that \eqref{AW sequence} holds for some $p\in\mathbb{C}$.
Then there exists a (unique) TD system with parameter array $\pi$ if and only if $\zeta_0=1$, $\zeta_{\delta}\ne 0$, and $\sum_{i=0}^{\delta}\zeta_i\prod_{\ell=i+1}^{\delta}(\theta_0-\theta_{\ell})(\theta_0^{\ster}-\theta_{\ell}^{\ster})\ne 0$.
\end{theorem}

\noindent
We remark that the left-hand side of the last condition on the $\zeta_i$ is a certain value of the \emph{Drinfel'd polynomial} of the corresponding TD system; cf.~\cite{IT2009JCISS,IT2010KJM}.
See, e.g., \cite{NT2009LAAa,NT2010LAAb,Bockting-Conrad2012LAA} for more results on TD systems.

Given the above progress in the theory of TD systems, it is important to `pull back' the results to the study of $Q$-polynomial distance-regular graphs.
For example, Pascasio \cite{Pascasio2002EJC} used the symmetric and unimodal property of the shape of $\Phi$ to study the multiplicities $m_i$ of a $Q$-polynomial distance-regular graph $\G$; cf.~Section \ref{sec: unimodality of mi}.
Terwilliger \cite{Terwilliger2005GC} `extended', so to speak, the split decompositions of the TD systems on the irreducible $\TT$-modules to the entire standard module $\mathbb{C}^v$, and obtained the \emph{split} and \emph{displacement decompositions for} $\G$.
Ito and Terwilliger \cite{IT2009EJC} used these decompositions to show that for the forms graphs there are four natural algebra homomorphisms from $U_q(\widehat{\mathfrak{sl}}_2)$ to $\TT$ via the so-called $q$-\emph{tetrahedron algebra}
$\boxtimes_q$ \cite{IT2007CA}, and that $\TT$ is generated by each of their images together with the center $Z(\TT)$.
Corresponding results for the case $p=2$, i.e., for Hamming and Doob graphs, were recently obtained by Morales and Pascasio \cite{MP2012pre}.
See also \cite{IT2009MMJ,Kim2009EJC,Kim2010DM} for more results on the split and displacement decompositions.
Worawannotai \cite{Worawannotai2012PhD} applied a similar idea to dual polar graphs to show (among other results) that there are two algebra homomorphisms from the quantum algebra $U_q(\mathfrak{sl}_2)$ to $\TT$, and that $\TT$ is again generated by each of their images together with $Z(\TT)$.
The split and displacement decompositions have also been applied to the Assmus-Mattson theorem for codes in $Q$-polynomial distance-regular graphs \cite{Tanaka2009EJC}; cf.~\cite[\S 2.8]{bcn}.
