


%\section{Terwilliger algebra and combinatorics}\label{sec:Talg6}

In this section, let $\G$ be a distance-regular graph with diameter $D\geq 3$, valency $k\geq 3$, and eigenvalues $k =
\theta_0 > \theta_1 > \cdots > \theta_D$. Concerning $1$-homogeneity of distance-regular graphs, we shall occasionally
consider the following weaker concepts. We say $\G$ is $1$-\emph{homogeneous with respect to an edge} $xy$ if the
parameters $p_{i,j;r,s}$ exist with respect to $x,y$ for all $i,j,r,s=0,1,\dots,D$; cf.~Section \ref{sec:homogeneity4}.
We say $\G$ is $1$-\emph{homogeneous with respect to a vertex} $x\in V$ if it is $1$-homogeneous with respect to the
edge $xy$ for every $y\in \G(x)$ and the parameters $p_{i,j;r,s}$ do not depend on the choice of $y$.


\subsection{Homogeneity and tight distance-regular graphs}

\subsubsection{Tight distance-regular graphs}
\label{sec:tightDRG}

Juri\v{s}i\'{c}, Koolen, and Terwilliger \cite{JKT2000JAC} showed the following so-called `fundamental bound':
\begin{equation}\label{FB}
	\left(\theta_1 + \frac{k}{a_1+1}\right) \left(\theta_D + \frac{k}{a_1+1}\right) \geq -\frac{ka_1b_1}{(a_1+1)^2}.
\end{equation}
For $a_1 =0$, equality holds if and only if $\G$ is bipartite.
One way to prove this bound is to use the fact
(\cite[Thm.~4.4.3]{bcn}) that $(\eta_i-\tilde{\theta}_1)(\eta_i-\tilde{\theta}_D)\leq 0$ for $i=2,3,\dots,k$, where
$\tilde{\theta}_1=-1-\frac{b_1}{1+\theta_1}$, $\tilde{\theta}_D=-1-\frac{b_1}{1+\theta_D}$, and
$a_1=\eta_1\geq \eta_2 \geq \dots \geq \eta_k$ are the eigenvalues of a local graph; cf.~\cite{JuKo00EuJC}. This
immediately shows that if $a_1\ne 0$ then equality holds if and only if every (or at least one) local graph is
connected strongly regular with non-trivial eigenvalues $\tilde{\theta}_1,\tilde{\theta}_D$.
We may also prove \eqref{FB} by considering the determinants of the Gram matrices of the three vectors
$E\textbf{e}_x,E\textbf{e}_y,E\chi_{1,1}(x,y)$ for adjacent vertices $x,y\in V$ and $E\in\{E_1,E_D\}$, where
$\chi_{1,1}(x,y)$ is the characteristic vector of $\G_{1,1}(x,y)=\G(x)\cap\G(y)$; cf.~\cite{JKT2000JAC}. See also
\cite{Pascasio2003DM} for another proof. We say $\G$ is {\em tight} if $a_1 \neq 0$ and equality holds in \eqref{FB}.
Juri\v{s}i\'{c} et al.~\cite{JKT2000JAC} showed that $\G$ is tight if and only if $a_1 \neq 0$, $a_D =0$,
and $\G$ is $1$-homogeneous.
To be more precise, call an edge $xy$ \emph{tight with respect to} a non-trivial
eigenvalue $\theta$ if $E\textbf{e}_x,E\textbf{e}_y,E\chi_{1,1}(x,y)$ are
linearly dependent, where $E$ is the primitive idempotent corresponding to
$\theta$. Then the following properties are all equivalent: (i) $\G$ is
tight; (ii) $a_1\ne 0$ and every (or at least one) edge of $\G$ is tight
with respect to both $\theta_1$ and $\theta_D$; (iii) $a_1\ne 0$, $a_D=0$, and
$\G$ is $1$-homogeneous (or $1$-homogeneous with respect to an edge).
Pascasio \cite{Pascasio2001GC} showed that if $\G$ is $Q$-polynomial then the following
properties are equivalent: (i) $\G$ is tight; (ii) $\G$ is non-bipartite and $a_D=0$; (iii) $\G$ is non-bipartite and
$a_D^{\ster}=0$. More characterizations of the tightness property will be given in the next
sections. The fundamental bound inspired quite a bit of the later research by Terwilliger and his students.

It follows from the above result of Pascasio that the non-bipartite antipodal $Q$-polynomial distance-regular graphs are tight; examples are the Johnson graph $J(2D,D)$, the
halved $2D$-cube, the non-bipartite Taylor graphs and the Meixner $2$-cover; cf.~Section \ref{sec:antipodalDRG}.
There are several sporadic examples known, all of which have diameter $4$, and of which only one is primitive, namely
the Patterson graph.
Juri\v{s}i\'{c} and Koolen \cite[Thm.~3.2]{JK2002DM} showed that tight distance-regular graphs with $D=3$ are precisely the non-bipartite Taylor graphs.
Suda \cite{Suda2012EJC} recently gave a simple proof of this result by looking at the
intersection matrix $L$; cf.~\eqref{matrixl}.

Using the fact that the Patterson graph, the Meixner $4$-cover, the $3.O_7(3)$-graph, and the $3.O_6^-(3)$-graph are
tight and hence $1$-homogeneous, one can easily show that the minimal convex subgraph of two vertices at distance two
is a complete multipartite graph $K_{n \times t}$ with $n \geq 2$, $t \geq 2$.
This leads in each of the cases to its
uniqueness as a distance-regular graph; cf.~\cite{JK2003JAC,JuKo07JAC,JuKo11,JuKopre,BrJuKo08}.

The family of tight antipodal distance-regular graphs with $D=4$ is called the $\mathrm{AT}4$-family. That they are
$1$-homogeneous gives rise to several feasibility conditions; cf.~\cite{JuKo00EuJC}.
Juri\v{s}i\'{c} and Koolen
\cite{JuKo11} classified the members of the $\mathrm{AT}4$-family with complete multipartite $\mu$-graphs. Juri\v{s}i\'{c},
Munemasa, and Tagami \cite{JuMuTa10} simplified, generalized, and strengthened some of the results in \cite{JuKo11}.

Vidali and Juri\v{s}i\'{c} \cite{VJ2013pre} recently showed the non-existence of primitive tight distance-regular graphs with classical parameters $(D,b,b-1,b^{D-1})$, where $D\geq 4$ and $b>1$.

\subsubsection{The \texorpdfstring{$\mathrm{CAB}$}{CAB} condition and \texorpdfstring{$1$}{1}-homogeneous distance-regular graphs}

Juri\v{s}i\'{c} and Koolen \cite{JuKo00DCC} introduced the $\mathrm{CAB}_j$ condition.
For vertices $x,y\in V$
at distance $i =0,1, \dots, D$, define the sets
$C_i(x,y) = \G_{i-1}(x)\cap \G(y)$, $A_i(x,y) =
\G_i(x) \cap \G(y)$, and $B_i(x,y) = \G_{i+1}(x) \cap \G(y)$ (with $\G_{-1}(x) = \G_{D+1}(x) =
\emptyset$).
For $j=0,1, \ldots, D$, we say $\G$ \emph{satisfies} $\mathrm{CAB}_j$, if for
all $i=0,1, \ldots, j$ and $x, y\in V$ at distance $i$, the partition $\{C_i(x,y),
A_i(x,y), B_i(x,y)\}$ of the local graph $\Upsilon(y)$ is equitable (where we
assume that empty sets are excluded from the partition).
It is clear that $\G$ satisfies $\mathrm{CAB}_0$, and that $\G$ satisfies $\mathrm{CAB}_1$ if and only if it is locally strongly regular.
Note that if $\G$ satisfies
$\mathrm{CAB}_2$ then the $\mu$-graph $\Upsilon(x,y)$ for vertices $x, y\in V$ at distance $2$ is
regular.
Juri\v{s}i\'{c} and Koolen \cite{JuKo00DCC} showed that if $\G$ satisfies $\mathrm{CAB}_j$ then for all $i=0,1,\dots,j$ and $x,y\in V$ at
distance $i$, the quotient matrix of $\{C_i(x,y),
A_i(x,y), B_i(x,y)\}$ does not depend on the pair $x,y$, but only on $i$.
They also showed that if $a_1 \neq 0$ then $\G$ satisfies $\mathrm{CAB}_D$ if and only if it is $1$-homogeneous.
Note that if $a_1=0$ then
$\G$ always satisfies $\mathrm{CAB}_D$.
Nomura \cite{N294} showed that  the $1$-homogeneous distance-regular graphs of order $(s,t)$ with $s \geq 2$, $t \geq
1$ are exactly the regular near $2D$-gons, a result that can be shown easily using the $\mathrm{CAB}_D$
condition.
Juri\v{s}i\'{c} and Koolen \cite{JuKo00DCC} also determined the $1$-homogenous Terwilliger graphs,
and gave an algorithm to determine all $1$-homogeneous distance-regular graphs
that are locally a given strongly regular graph.
See also \cite{JK2003JAC}.


\subsubsection{More results on homogeneity}\label{sec:homogeneity}
Miklavi\v{c} \cite{Miklavic2004EJC} showed that the triangle-free $Q$-polynomial distance-regular graphs are $1$-homogeneous.
Note that if $a_1=0$ then the multiplicity of an eigenvalue distinct from $\pm k$ is at least $k$ by Terwilliger's tree bound; cf.~Section \ref{sec:tree_bound}.
Coolsaet, Juri\v{s}i\'{c}, and Koolen \cite{CoJuKo08EuJC} showed among other results that $\G$ is $1$-homogeneous if it has an eigenvalue with multiplicity $k$, $a_1=0$, $a_2\ne 0$, and $a_4=0$ (when $D\geq 4$), and then ruled out the infinite family of intersection arrays $\{2\mu^2 + \mu, 2\mu^2 + \mu -1, \mu^2, \mu, 1; 1, \mu, \mu^2, 2\mu^2 + \mu -1, 2\mu^2 + \mu\}$ ($\mu\geq 2$).
For $\mu=1$, this intersection array is uniquely realized by the dodecahedron.
Juri\v{s}i\'{c}, Koolen, and \v{Z}itnik \cite{JKZ2008EJC} showed among other results that if $\G$ is primitive and has an eigenvalue with multiplicity $k$, $a_1=0$, and $D=3$, then the association scheme underlying $\G$ is formally self-dual and thus $\G$ is $Q$-polynomial and $1$-homogeneous.

%Concerning $2$-homogeneity, 
Nomura \cite{Nomura1995JCTB} classified the $2$-homogeneous bipartite distance-regular graphs; cf.~Section \ref{sec:antipodalDRG}.
Nomura \cite{Nomura1996P} also classified the $2$-homogeneous generalized odd graphs.
Yamazaki \cite{Yamazaki1996JCTB} observed that if $\G$ is bipartite then $\G$ has an eigenvalue with multiplicity $k$ if and only if it is $2$-homogeneous, while Curtin \cite{Curtin1998DM} showed that if $\G$ is bipartite then $\G$ is $2$-homogeneous if and only if it is $Q$-polynomial and antipodal.



\subsection{Thin modules}
\label{sec:thinness}

Thin irreducible $\TT$-modules with endpoint $1$ have been extensively
studied;
see e.g., \cite{Terwilliger1993N,GT2002EJC,Terwilliger2002LAA,Terwilliger2004JAC} and Section \ref{sec:thinness-Q}. For
example, let $\mathbf{v}$ be a nonzero vector in $E_1^{\ster}\mathbb{C}^v$ which is orthogonal to $A_1\mathbf{e}_x$, so
that $E_0\mathbf{v}=0$. Go and Terwilliger \cite{GT2002EJC} showed that if $E_i\mathbf{v}$ vanishes for some
$i=1,2,\dots,D$ then $i\in\{1,D\}$ and $\AL\mathbf{v}$ is a thin irreducible $\TT$-module with endpoint $1$ and
diameter $D-2$. There is also a characterization of thin irreducible $\TT$-modules with endpoint $1$, involving the
pseudo primitive idempotents introduced by Terwilliger and Weng \cite{TW04}. Let $\theta\in\mathbb{C}$ (not necessarily
an eigenvalue of $\G$).
The \emph{pseudo cosine sequence for} $\theta$ is the sequence $(\sigma_i)_{i=0}^D$ defined by
$\sigma_0=1$ and the recursion $c_i\sigma_{i-1}+a_i\sigma_i+b_i\sigma_{i+1}=\theta\sigma_i$ for $i=0,1,\dots,D-1$;
cf.~\eqref{standard_sequence}.
A \emph{pseudo primitive idempotent} $E_{\theta}$ \emph{associated with} $\theta$ is then any
nonzero scalar multiple of $\sum_{i=0}^D\sigma_iA_i$.
We also define $E_{\infty}$ to be any  nonzero scalar multiple of $A_D$. Let $\mathbf{v}$ be as
above, and let $(\AL;\mathbf{v})=\{M\in\AL:M\mathbf{v}\in E_D^{\ster}\mathbb{C}^v\}$.
Note that $J\in (\AL;\mathbf{v})$.
Terwilliger and Weng \cite{TW04} showed that $\TT\mathbf{v}$ is a thin irreducible
$\TT$-module (with endpoint $1$) if and only if $\dim(\AL;\mathbf{v})\geq 2$. Moreover, if this is the case, then
$\dim(\AL;\mathbf{v})=2$ and we have $(\AL;\mathbf{v})=\mathrm{span}_{\mathbb{C}}\{J,E_{\tilde{\eta}}\}$, where $\eta$
is the local eigenvalue corresponding to $\TT\mathbf{v}$ and
\begin{equation*}
	\tilde{\eta}=\begin{cases} \infty & \text{if}\ \eta=-1, \\ -1 & \text{if}\ \eta=\infty, \\ -1-\frac{b_1}{1+\eta} & \text{if}\ \eta\ne -1,\infty. \end{cases}
\end{equation*}
Terwilliger \cite{Terwilliger2004JAC} obtained an
inequality\footnote{\label{similar to SET}We may remark that the
discussions in \cite{Terwilliger2004JAC} and those in the proof of
the spectral excess theorem (Theorem \ref{spectral excess theorem})
given by Fiol and Garriga \cite{FG97} are similar in nature. In
\cite{Terwilliger2004JAC}, Terwilliger is concerned with the
thinness of irreducible $\TT$-modules with endpoint $1$ of a
distance-regular graph, whereas Fiol and Garriga \cite{FG97} take a
``local approach'', which can be understood as being concerned with
the thinness of the primary $\TT$-module of a general (finite,
simple, and connected) graph. (See \cite{Terwilliger1993N} for the
basic theory about the Terwilliger algebra of a general graph.) In
both cases, the characterization of the thinness as equality in a
bound is obtained by focusing on two specific vectors in
$E_D^{\ster}\mathbb{C}^v$.} involving the local eigenvalues of
$\G$, and showed that equality is attained if and only if $\G$ is
$1$-thin with respect to the base vertex $x$. Go and Terwilliger
\cite[Thm.~13.7]{GT2002EJC} showed that the following properties
are all equivalent: (i) $\G$ is tight; (ii) $\G$ is non-bipartite,
$a_D=0$, and $\G$ is $1$-thin; (iii) $\G$ is non-bipartite,
$a_D=0$, and $\G$ is $1$-thin with respect to at least one vertex.

We have some comments. It is well known that $a_1\ne 0$ implies $a_i\ne 0$ for
$i=1,2,\dots,D-1$; cf.~\cite[Prop.~5.5.1]{bcn}.
Dickie and Terwilliger \cite{DT1998JAC} showed that if $\G$ is $1$-thin with respect to at least one vertex
then $a_1=0$ implies $a_i=0$ for $i=1,2,\dots,D-1$. We note that these results
have dual versions for $Q$-polynomial association schemes;
cf.~\cite{Dickie1995D,DT1998JAC}.

Collins \cite{Collins1997GC} showed that $\G$ is thin with $c_3=1$ if and only if it is a generalized octagon of order
$(1,t)$. This shows that if $\G$ is thin then the numerical girth $g$ is at most $8$ (and cannot be $7$).
(Collins \cite{Collins1997GC} only mentioned the implication for the girth of $\G$.)
Suzuki \cite{Suzuki2006EJC} strengthened this result as follows.
Suppose $\G$ is of order $(s,t)$, and recall that $g$ coincides with the geometric girth in this case.
Suzuki showed among other
results that (i) $g\leq 11$ if there is a thin irreducible $\TT$-module with endpoint $3$; (ii) $\G$ is a regular near
polygon\footnote{The referee kindly pointed out an error in \cite[Thm.~1.2(ii)]{Suzuki2006EJC}.} if and only if it is $1$-thin; (iii) if $g\geq 8$ then $\G$ is a generalized $2D$-gon of order $(1,t)$ if and
only if it is $1$- and $2$-thin; (iv) if $g\geq 8$ then $\G$ is a generalized octagon of order $(1,t)$ if and only if
it is $1$-, $2$-, and $3$-thin.

Curtin \cite{Curtin1999GC} studied the Terwilliger algebras of bipartite distance-regular graphs.
Suppose for the moment that $\G$ is bipartite.
Then he showed among other results that $\G$ is always $1$-thin with a unique irreducible $\TT$-module with endpoint $1$ up to isomorphism, and that if $\G$ is $2$-thin with respect to the base vertex $x$ then the intersection array is determined by $D$ and the multiplicity in $\mathbb{C}^v$ of each of the irreducible $\TT$-modules $W$ with endpoint $2$, together with the scalar $\psi(W)=-\frac{b_2b_3}{c_2(\eta(W)+1)}-1$, where $\eta(W)$ is the eigenvalue of $E_2^*A_2E_2^*$ on $E_2^*W$, which is an eigenvalue of the local graph of $x$ in the halved graph of $\G$.
See also \cite{Curtin1999EJC}.
In particular, if $\G$ is $2$-thin with respect to $x$ with (at most) two irreducible $\TT$-modules $W_1,W_2$ with endpoint $2$ up to isomorphism, then it turns out that the intersection array is determined by $D,k,c_2,\psi(W_1)$, and $\psi(W_2)$.

Collins \cite{Collins2000DM} studied in detail the relation between the irreducible $\TT$-modules of an almost bipartite distance-regular graph $\G$ and those of its bipartite double $\tilde{\G}$.
In particular, he showed that $\G$ is thin if and only if $\tilde{\G}$ is thin.



\subsection{Vanishing Krein parameters}\label{sec:vanishingKrein}

Vanishing of Krein parameters often leads to strong (combinatorial) consequences.
A classical example is a result of Cameron, Goethals, and Seidel \cite{CGS1978JA} which states that if a strongly regular graph satisfies either $q_{11}^1=0$ or $q_{22}^2=0$ then for every vertex, the induced subgraphs on both of the subconstituents are strongly regular.\footnote{In passing, by the results of \cite{CGS1978JA} one can quickly find all the irreducible $\TT$-modules of a strongly regular graph.
In particular, it is always thin; cf.~\cite{TY1994KJM}.}
See \cite{Godsil1992AJC,JK2002DM,Jurisic2003DM} for similar results for antipodal distance-regular graphs with diameter $3$ or $4$.
In this section, we discuss more results on this topic.


\subsubsection{Triple intersection numbers}\label{sec:triple intersection numbers}
Coolsaet, Juri\v{s}i\'{c}, and others used
vanishing Krein parameters to obtain information on triple intersection
numbers as follows.
Let $x, y, z\in V$.
For $r, s, t =0,1,\dots, D$, let $p^{x,y,z}_{r, s, t} = |\{ u\in V : d(x,u) = r, d(y, u) =
s, d(z,u) = t\}|$.
Now, if $q^h_ {ij} =0$ then it follows from \eqref{3-tensor} that
\begin{equation}\label{triples}
	\sum_{r, s, t = 0}^D Q_{ri} Q_{sj} Q_{th} p^{x,y,z}_{r, s, t} = 0.
\end{equation}
This equation gives some extra information on the triple intersection numbers. We note that
\eqref{3-tensor} was also used earlier by Terwilliger \cite{Terwilliger1985AGG} to study the number of $4$-vertex
configurations with given mutual distances; he showed that if $\G$ is $Q$-polynomial then such numbers can be computed
from the intersection array and the numbers of $4$-vertex cliques in $\G_1,\G_2,\dots,\G_{\lfloor D/2\rfloor}$.
Using \eqref{triples}, Coolsaet and Juri\v{s}i\'{c} \cite{CoJu08} ruled out the infinite family of intersection arrays
$\{4r^3+8r^2+6r+1, 2r(r+1)(2r+1), 2r^2+2r+1; 1, 2r(r+1), (2r+1)(2r^2+2r+1)\}$ $(r \geq 2)$.
The case $r=1$, i.e., $\{19, 12, 5; 1, 4, 15\}$, was eliminated by Neumaier; cf.~\cite[\S 5.5A]{BCNcoradd}.
Coolsaet and Juri\v{s}i\'{c} also
ruled out the intersection array $\{ 74, 54, 15; 1, 9, 60\}$.
Juri\v{s}i\'{c} and Vidali \cite{JurVidpre}
used the above idea of triple intersection numbers to show that there exists a set of vertices mutually at distance
$3$ of size $p^3_{33}+2$  for distance-regular graphs with intersection arrays $\{(2r^2-1)(2r+1), 4r(r^2-1), 2r^2; 1,
2(r^2-1), r(4r^2-2)\}$ or $\{ 2r^2(2r+1), (2r-1)(2r^2+r+1), 2r^2; 1, 2r^2, r(4r^2-1)\}$ $( r \geq 2)$, and showed that
consequently such graphs do not exist. Urlep \cite{urlep12} used \eqref{triples} to rule out the intersection arrays
$\{(r + 1)(r^3 - 1), r(r - 1)(r^2 + r - 1), r^2 - 1; 1, r(r + 1), (r^2 - 1)(r^2 + r - 1)\}$ ($r \geq 3$). For $r=2$,
this intersection array is uniquely realized by the halved $7$-cube.
Vidali \cite{Vidali} recently used \eqref{triples} again to rule out the intersection array $\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\}$.


\subsubsection{Hadamard products of two primitive idempotents}
\label{sec: Hadamard products} Another important use of vanishing Krein parameters is the study of pairs of non-trivial
primitive idempotents $E,F$ such that $E\circ F$ is a linear combination of a small number of primitive idempotents;
cf.~\eqref{Krein}.
For convenience, we define
$e(M)=\{E_i:ME_i\ne 0\}$ for $M\in\AL$. Pascasio \cite{Pascasio1999JAC} showed that non-trivial primitive idempotents
$E,F$ satisfy $|e(E\circ F)|=1$ precisely when one of the following holds: (i) $\G$ is tight, $\{E,F\}=\{E_1,E_D\}$,
and $e(E\circ F)=\{E_{D-1}\}$; (ii) $\G$ is bipartite and $E_D\in\{E,F\}$.

Suppose for the moment that $\G$ is bipartite with $D\geq 4$.
Let $\theta,\theta'$ be eigenvalues of $\G$ other than $\pm k$, and let $E,F$ be the primitive idempotents associated with $\theta,\theta'$.
Then $|e(E\circ F)|>1$ by (ii) above.
MacLean \cite{MacLean2000DM} called the pair $\{E,F\}$ \emph{taut} if $|e(E\circ F)|=2$.
He showed that $|e(E\circ F)|=2$ if and only if $\theta,\theta'$ attain equality in what he called the `bipartite fundamental bound'.
We comment on the proof of this result. Let $E=E_i$ and $F=E_j$, and for $t=0,1,2$, let $\mathbf{f}_t$ be the vector in
$\mathbb{R}^{D+1}$ with $h$-coordinate
\begin{equation}\label{MacLean's vector}
	\theta_h^t\sqrt{\frac{q_{ij}^hm_h}{m_im_j}} \quad (h=0,1,\dots,D).
\end{equation}
Then $\mathbf{f}_0,\mathbf{f}_1,\mathbf{f}_2$ are linearly dependent if and only if $|e(E\circ F)|=2$, and computing the determinant of the (positive semidefinite) Gram matrix of $\mathbf{f}_0,\mathbf{f}_1,\mathbf{f}_2$ yields the bipartite fundamental bound.
See \cite{MT2006DM,MacLean2012DM} for more proofs of this bound.
We say $\G$ is \emph{taut} if it has a taut pair of primitive idempotents and is not $2$-homogeneous.
MacLean \cite[Thm.~1.4]{MacLean2000DM} showed that $\{E,F\}$ is taut precisely when one of the following holds:
(i) $\G$ is taut and $\{E,F\}\in\{\{E_h,E_{\ell}\}:h\in\{1,D-1\},\ell\in\{\tau,D-\tau\}\}$ where $\tau=\lfloor D/2\rfloor$; (ii) $\G$ is $2$-homogeneous and $\{E,F\}\cap\{E_1,E_{D-1}\}\ne\emptyset$.
For $D=4,5$, $\G$ is taut or $2$-homogeneous if and only if $\G$ is antipodal \cite[\S\S7--8]{MacLean2000DM}.
MacLean and Terwilliger \cite{MT2006DM} showed among other results that if $D$ is odd then the following are equivalent:
(i) $\G$ is taut or $2$-homogeneous; (ii) $\G$ is antipodal and $2$-thin; (iii) $\G$ is antipodal and $2$-thin with respect to at least one vertex; see also \cite{MT2008DM}.
Examples of taut graphs with odd $D\geq 5$ are the Doubled Odd graphs, the Doubled Hoffman-Singleton
graph, the Doubled Gewirtz graph, and the Doubled $77$-graph; cf.~ \cite[p.~131]{MacLean2003JAC}. For $D$ even and at least
$6$, MacLean \cite[Thm.~5.8]{MacLean2000DM} showed that $\G$ is taut or $2$-homogeneous if and only if its halved
graphs are tight. If $\G$ is taut in this case, then it turns out however that $D\ne 6$ and that no known example of a
tight distance-regular graph with diameter at least $4$ can be a halved graph of $\G$; cf.~\cite{MacLean2004JCTB}.

Retaining the situation of the last paragraph, let $\Delta_E$ be the representation diagram\footnote{The \emph{representation diagram} of $E=E_i$ is
the simple graph with vertex set $\{0,1,\dots,D\}$, where two distinct vertices $h,\ell$ are adjacent whenever
$q_{ih}^{\ell}\ne 0$.} of $E=E_i$, and let $(u_h)_{h=0}^D$ be the standard sequence associated with $E$. Note that $0$ and
$D$ are leaves (i.e., terminal vertices) in $\Delta_E$, and that $j$ is a leaf in $\Delta_E$ precisely when $|e(E\circ
F)|=2$ and $F\in e(E\circ F)$. Lang \cite{Lang2004JCTB} showed that
\begin{equation}\label{Lang's inequality}
	(u_1-u_{h+1})(u_1-u_{h-1})\geq (u_2-u_h)(u_0-u_h) \quad (h=1,2,\dots,D-1),
\end{equation}
with equality for every $h=1,2,\dots,D-1$ (or just for $h=3$) if and only if $u_{h-1}-pu_h+u_{h+1}$ is independent of
$h=1,2,\dots,D-1$ for some $p\in\mathbb{R}$. When $E$ attains equality, Lang \cite{Lang2002EJC} showed that (i) $u_D\ne
1$ if and only if $\Delta_E$ is a path (i.e., $E$ is $Q$-polynomial); (ii) $u_D=1$ if and only if $\Delta_E$ is the
disjoint union of two paths. It follows that if case (ii) occurs then $\G$ is antipodal and the folded graph is
$Q$-polynomial; cf.~\cite[Thm.~10.2, 10.4]{Lang2004JCTB}. Note that in both cases (i) and (ii), $E$ is a \emph{tail},
i.e., $|e(E\circ E)|\leq 3$ and $|e(E\circ E)\backslash\{E_0,E\}|\leq 1$. Conversely, Lang \cite{Lang2002EJC} showed
that if $E$ is a tail and $D\ne 6$ then $E$ attains equality in \eqref{Lang's inequality}.
Lang \cite{Lang2003JAC} also showed that $\Delta_E$ has a leaf other than $0,D$ if and only if $E$ attains
equality in \eqref{Lang's inequality} and case (ii) occurs above. One of the other results in \cite{Lang2004JCTB} is
that if $D\geq 6$ and $\G$ has more than one primitive idempotent that attains equality in \eqref{Lang's inequality},
then $\G$ is the $D$-cube.

Suppose now that $\G$ is arbitrary (i.e., $D\geq 3$ and not necessarily bipartite).
By considering the Gram matrix of $\mathbf{f}_0$ and $\mathbf{f}_1$, Pascasio \cite{Pascasio2003DM} later extended some of the results in \cite{Pascasio1999JAC}, as well as the
fundamental bound, to the level of $P$-polynomial character algebras.
Tomiyama \cite{Tomiyama2001DM} considered the situation where one of $1,D$ is a leaf in $\Delta_E$ and generalized some of the results in \cite{Lang2003JAC,Lang2004JCTB,Pascasio1999JAC,Pascasio2001GC}.

Assume $E=F$ (so $i=j$) and $\theta(=\theta')\neq\pm k$.
Then $|e(E\circ E)|\geq 2$.
We call $E$ a \emph{light tail}\footnote{It is easy to see that if $e ( E \circ E ) = \{ E_0, E \}$ then $\G$ is antipodal with $D=3$; cf.~\cite[Thm.~ 4.1(b)]{JTZ2010EJC}. We view this case as degenerate,
so we propose to assume $E \not\in e ( E \circ E )$ as well in the definition of a light tail.} \cite{JTZ2010EJC} if $|e(E\circ E)|=2$.
Let $\mathbf{f}_0',\mathbf{f}_1'$ be the vectors obtained from $\mathbf{f}_0,\mathbf{f}_1$, respectively, by the removal of
the $0$-coordinate. Note that $\mathbf{f}_0',\mathbf{f}_1'$ are linearly dependent if and only if $E$ is a light tail.
Juri\v{s}i\'{c}, Terwilliger, and \v{Z}itnik \cite{JTZ2010EJC} considered the Gram matrix of
$\mathbf{f}_0',\mathbf{f}_1'$. In this case, the resulting inequality gives a lower bound on the multiplicity of
$\theta$; cf.~Proposition \ref{light tail bound}. They showed among other results that distance-regular graphs with a
light tail are close to being $1$-homogeneous, i.e., the parameters $p_{i,j;r,s}$ exist with respect to, and are
independent of, every pair of adjacent vertices $x,y\in V$ for all $i,j,r,s=0,1,\dots,D$ except possibly $i=j=2,3,\dots,D-1$. In
particular, the local graphs are strongly regular. We note that these results generalize those of Cameron, Goethals,
and Seidel \cite{CGS1978JA} mentioned at the beginning of Section \ref{sec:vanishingKrein}. They indeed showed that
primitive strongly regular graphs with a light tail (and $k\geq 3$) are precisely the Smith graphs.


\subsection{Relaxations of homogeneity}

In the previous sections, we explored connections among homogeneity, thin modules, tightness,
local graphs, Hadamard products of two primitive idempotents, and so on. In fact, many of these results can be
generalized in several directions, as we discuss below.

We say $\G$ is \emph{pseudo $1$-homogeneous with respect to an edge} $xy$ \cite{JT2008JAC} if the parameters $p_{i,j;r,s}$ exist with respect to $x,y$ for all $i,j,r,s=0,1,\dots,D$ except possibly $i=j=D$.
Let $\theta\in\mathbb{R}\backslash \{k\}$, and let $E_{\theta}$ be a pseudo primitive idempotent associated with $\theta$; cf.~Section \ref{sec:thinness}.
We say the edge $xy$ is \emph{tight with respect to} $\theta$ \cite{JT2008JAC} if a non-trivial linear combination of
$E_{\theta}\mathbf{e}_x,E_{\theta}\mathbf{e}_y,E_{\theta}\chi_{1,1}(x,y)$ is contained in the subspace
$\mathrm{span}_{\mathbb{R}}\{\mathbf{e}_z:z\in \G_{D,D}(x,y)\}$. Juri\v{s}i\'{c} and Terwilliger \cite{JT2008JAC}
showed among other results that if $a_1\ne 0$ then the edge $xy$ is tight with respect to two distinct real numbers if
and only if $\G$ is pseudo $1$-homogeneous with respect to $xy$ and the induced subgraph on $\G_{1,1}(x,y)$ is not a
clique. Under the condition $a_1\ne 0$, Curtin and Nomura \cite{CN2005JCTB} characterized the
situation where $\G$ is $1$-thin with respect to $x$ with precisely two non-isomorphic irreducible $\TT(x)$-modules with
endpoint one, in terms of the pseudo $1$-homogeneous property\footnote{It should be
remarked that Curtin and Nomura \cite{CN2005JCTB} do not require the existence of the parameter $p_{D,D-1;r,s}$ with
respect to $x,y$.} of the edges $xy$ ($y\in\G(x)$).
They studied in detail the case where $a_1=0$ as well.
Extending the work of Pascasio \cite{Pascasio1999JAC} on the tightness property, Pascasio and Terwilliger \cite{PT2006LAA} described exactly when $E_{\theta}\circ E_{\theta'}$ with $\theta,\theta'\in\mathbb{R}$ is a scalar multiple of $E_{\tau}$ for some $\tau\in\mathbb{R}$.

Suppose for the moment that $\G$ is bipartite with $D\geq 4$, and let $x,y$ be vertices with $d(x,y)=2$.
Curtin \cite[\S\S 4--5]{Curtin1998DM} showed that $\G$ is $2$-homogeneous if and only if $|\G_{1,1}(x,y)\cap\G_{i-1}(z)|$ depends only on $i=1,2,\dots,D-1$ and is independent of $z\in \G_{i,i}(x,y)$.
We say $\G$ is \emph{almost $2$-homogeneous} \cite{Curtin2000EJC} if the same condition holds for $i=1,2,\dots,D-2$.
Recall that $\G$ is $1$-thin with a unique irreducible $\TT$-module with endpoint $1$ up to
isomorphism.
Curtin \cite{Curtin2000EJC} showed among other results that $\G$ is almost $2$-homogeneous if and only if it
is $2$-thin with a unique irreducible $\TT$-module with endpoint $2$ up to isomorphism.
Curtin \cite{Curtin2000EJC}
and Juri\v{s}i\'{c}, Koolen, and Miklavi\v{c} \cite{JKM2005JCTB} classified the almost $2$-homogeneous bipartite
distance-regular graphs: the $2$-homogeneous graphs (cf.~Section \ref{sec:antipodalDRG}), the generalized
$2D$-gons of order $(1,k-1)$, the folded $2D$-cube, and the coset graph of the extended binary Golay
code.\footnote{See also \cite{Suzuki2008GC} for a generalization of this result (as well as Nomura's
classification \cite{Nomura1995JCTB,Nomura1996P} of bipartite or almost bipartite $2$-homogeneous distance-regular
graphs) to triangle-free distance-regular graphs. That the coset graph of the extended binary Golay code is almost
$2$-homogeneous was pointed out by Lang \cite[Lemma~3.4]{Lang2008EJC}.}
Lang \cite{Lang2008EJC} considered when
$E_{\theta}\circ E_{\theta}$ with $\theta\in\mathbb{C}\backslash\{k,-k\}$ is a linear combination of $J$ and $E_{\tau}$ for some $\tau\in\mathbb{C}$, and showed that this occurs precisely when $\G$ is almost $2$-homogeneous and $c_2\geq 2$.

In some cases, it is possible to get an equitable partition of
$V$ from $\Pi=\{\G_{i,j}(x,y):\G_{i,j}(x,y)\ne\emptyset,\,
i,j=0,1,\dots,D\}$, where $d(x,y)=h\in\{1,2\}$, by refining some of the
$\G_{i,i}(x,y)$ $(i=2,3,\dots,D)$ into two cells, even when $\Pi$ itself is
not equitable. This was worked out in detail by Miklavi\v{c} for
distance-regular graphs having classical parameters with $b<-1$, $a_1\ne 0$
\cite{Miklavic2005JCTB} ($h=1$), for bipartite $Q$-polynomial distance-regular
graphs with $c_2=1$ \cite{Miklavic2007DM} ($h=2$), and for the bipartite dual
polar graphs $\D_D(q)$ \cite{Miklavic2013GC} ($h=2$). See also
\cite{Miklavic2007EJC} for a description of the $\AL$-module spanned by
$\{\chi_{i,j}(x,y):i,j=0,1,\dots,D\}$ with $h=2$ (cf.~Section
\ref{sec:Qpolcharacterizations}) for bipartite $Q$-polynomial distance-regular
graphs. The parameters of the new equitable partition give rise to additional
integrality conditions, and he used these conditions to show that there is no
bipartite $Q$-polynomial distance-regular graph with $D=4$ and girth $6$;
cf.~Section \ref{sec:girth}.
