

%\section{Miscellaneous}\label{sec:misc}

\subsection{Distance-transitive graphs}\label{sec:dtgmisc} As
already expressed in `BCN' \cite[Ch.~7]{bcn}, it seems to be feasible to classify all distance-transitive graphs,
starting with the primitive ones, given the classification of finite simple groups. For more information on this, we
refer to the historical essay
--- and then state of the art survey --- by Ivanov \cite{IvanovDTG}, the
introduction to the field --- and survey --- by Cohen \cite{cohen04}, and the
(currently) most recent survey by Van Bon \cite{vanbon07}. Concerning the
classification of imprimitive distance-transitive graphs, we mention the
classification of antipodal distance-transitive covers of complete graphs by
Godsil, Liebler, and Praeger \cite{GLP98} and the classification of
distance-transitive covers of complete bipartite graphs by Ivanov, Liebler,
Penttila, and Praeger \cite{ILPP97}. Moreover, Alfuraidan and Hall \cite{AH09}
`finished' the classification of distance-regular graphs whose so-called
primitive core (i.e., the primitive graph obtained after halving and/or
quotienting) is a known distance-transitive graph with diameter at least three.



\subsection{The metric dimension}\label{sec:metricdim}

Given a graph, a {\em resolving set} $W$ is a set of vertices such
that every vertex in the graph is uniquely determined by the
distances to the vertices in $W$. The {\em metric dimension} of a
graph is the size of a smallest resolving set. Babai
\cite{Babaimetric} studied the metric dimension of graphs motivated
by the graph isomorphism problem (he actually studied the problem
more generally in coherent configurations). His results imply an
upper bound on the metric dimension for primitive distance-regular
graphs in terms of the number of vertices, the diameter, and the
valencies. Chv\'{a}tal \cite{Chvatalmetric} obtained an asymptotic
result on the metric dimension of the Hamming graphs, as a result
of his work on strategies for the game {\em Mastermind}. For
details, we refer to the survey paper by Bailey and Cameron
\cite{Metricdimension}. For recent results on the metric dimension
of Johnson graphs, Grassmann graphs, bilinear forms graphs, and
symplectic dual polar graphs, we refer to \cite{Johnsonmetric,
GWL13}, \cite{BaileyMeagher}, \cite{FengWang}, and \cite{GWL13DP},
respectively. Guo, Wang, and Li \cite{GWL13} also obtained results
on the Doubled Odd graphs, Doubled Grassmann graphs, and twisted
Grassmann graphs. The metric dimension of all `small'
distance-regular graphs was determined by Bailey
\cite{Baileysmall}. Bailey \cite{Baileyimprimitive} also related
the metric dimension of several families of imprimitive
distance-regular graphs to the metric dimension of corresponding
primitive distance-regular graphs. The {\em fractional metric
dimension} of vertex-transitive distance-regular graphs, in
particular Hamming and Johnson graphs, was studied by Feng, Lv, and
Wang \cite{FLWfractionalmetric}.


\subsection{The chromatic number}\label{sec:chromatic}

The {\em chromatic number} of a graph is the smallest number of
colors needed to color the vertices such that adjacent vertices
have different colors. Bipartite graphs clearly have chromatic
number $2$. It is not hard to see that the chromatic number of the
Hamming graph $H(d,q)$ equals $q$. Blokhuis, Brouwer, and Haemers
\cite{BlokhuisBH07} studied distance-regular graphs with chromatic
number $3$. They showed that besides the complete tripartite
graphs, the intersection number $a_1$ is at most $1$ in such
graphs, and they obtained several results for the case $a_1=1$; it
seems that the triangle-free case is much more difficult. All
graphs with chromatic number $3$ among the known distance-regular
graphs were classified by Blokhuis et al.~\cite{BlokhuisBH07}:
these are the complete tripartite graphs, the odd cycles, the Odd
graphs, the Hamming graphs $H(D,3)$, and nine exceptional graphs.
It was also shown that the folded cubes have chromatic number $4$.
Koolen and Qiao \cite{KQ15} classified the non-bipartite distance-regular
graphs with diameter three, valency $k$, and smallest eigenvalue at
most $-k/2$. Using these results, they obtained a complete
classification of the distance-regular graphs with diameter three
and chromatic number $3$. Hahn, Kratochv\'{\i}l,
\v{S}ir\'{a}\v{n}, and Sotteau
\cite{injectivecolor} obtained results on the chromatic number of the halved cubes; see also \cite{webchromatic}.
Etzion and Bitan \cite{colorJohnson} and Brouwer and Etzion \cite{BrEt11} provide a summary of results on the chromatic
number of the Johnson graphs.

\subsection{Cores}\label{sec:cores}

A \emph{core} is a graph having no endomorphisms other than
automorphisms. Every graph is homomorphically equivalent (i.e.,
there are homomorphisms in both directions) to a unique core,
called the \emph{core of} the graph. We say that a graph is
\emph{core-complete} if it is either a core or has a complete core.
Cameron and Kazanidis \cite{CK2008JAMS} showed among other results
that rank $3$ graphs are core-complete. Godsil and Royle
\cite{GR2011AC} showed among other results the core-completeness of
many infinite families of geometric strongly regular graphs. In
particular, by virtue of a result of Neumaier \cite{Neu80}, it
follows that, for given $m\geq 2$, all but finitely many strongly
regular graphs with smallest eigenvalue at least $-m$ are
core-complete. Roberson \cite{Rob16} finally showed
that all strongly regular graphs are core-complete. Concerning
general distance-regular graphs, Godsil and Royle
\cite{GR2011AC} showed that distance-transitive graphs are
core-complete, and that triangle-free non-bipartite
distance-regular graphs are cores. Huang, Lv, and Wang
\cite{HLW2014pre} studied cores and endomorphisms of the Grassmann
graphs.

\subsection{Modular representations}

Some work has been done on the adjacency algebra (denoted $\AL_K\subset M_{v\times v}(K)$) of a
distance-regular graph $\G$ over a field $K$ of characteristic $p>0$. Arad, Fisman, and Muzychuk
\cite[Thm.~1.1]{AFM1999IJM} showed among other results that $\AL_K$ is semisimple if and only if the
\emph{Frame number} $v^{D+1}\prod_{i=1}^D(k_i/m_i)$ (which is an integer) is not divisible by $p$. See also
\cite[Thm.~4.2]{Hanaki2000JA}. Hanaki \cite{Hanaki2002AM} showed that $\AL_K$ is a local algebra if $v$ is a
power of $p$.
For strongly regular graphs, Hanaki and Yoshikawa \cite{HY2005JAC} determined the structure of $\AL_K$ and studied the modular standard module $K^v$.
In this case, the $p$-rank of $M\in\AL_K$ (cf.~Section \ref{sec:prank}) can be interpreted as the dimension of the submodule $MK^v$, and their results provide us information as to which elements of $\AL_K$ we should look at.
Yoshikawa \cite{Yoshikawa2004JAC} determined the structure of $\AL_K$ for Hamming graphs. The
structure of $\AL_K$ for Johnson graphs was studied by Shimabukuro
\cite{Shimabukuro2005AC,Shimabukuro2011DM}. Shimabukuro \cite{Shimabukuro2007EJC} also computed the number of
irreducible representations of $\AL_K$ for the classical families of distance-regular graphs.
Shimabukuro and Yoshikawa \cite{SY2014pre} recently studied the structure of $\AL_K$ for Grassmann graphs.
For more information on modular representations of general (non-commutative) association schemes (i.e., homogeneous coherent configurations), we refer to the recent survey by Hanaki \cite{Hanaki2009EJC}.


\subsection{Asymptotic spectral analysis}\label{sec:spectral analysis}

Let $\G$ be a distance-regular graph with adjacency matrix $A$.
Observe that the complex Bose-Mesner algebra $\AL=\AL_{\mathbb{C}}$ of $\G$ is a commutative $*$-algebra, and that $\frac{1}{v}\tr$ is a state on $\AL$, i.e., a unital positive linear $*$-functional on $\AL$.
Thus, $(\AL,\frac{1}{v}\tr)$ is a classical algebraic probability space, and we may view $A$ as an algebraic random variable.
From this point of view, Hora \cite{Hora1998IDAQPRT} obtained, as variations of the central limit theorem, asymptotic spectral distributions for the families of Hamming graphs, Johnson graphs, halved cubes, and Grassmann graphs.
He used the information on the spectra of these graphs directly,
but then the method of quantum decomposition, first introduced in this context by Hashimoto \cite{Hashimoto2001IDAQPRT}, was applied to Hamming graphs by Hashimoto, Obata, and Tabei \cite{HOT2001P} and to Johnson graphs (among others) by Hashimoto, Hora, and Obata \cite{HHO2003JMP}, which provided a more conceptual and succinct (and `fully-quantum') approach to the results of Hora \cite{Hora1998IDAQPRT}.
See also \cite{HO2003P}.
Their theory, in the final form given in \cite{HO2007B,HO2008TAMS}, turns out to be closely related to the Terwilliger algebra (in the case of distance-regular graphs; though they did not use the language of the Terwilliger algebra).
Let $\TT$ be the Terwilliger algebra of $\G$ with respect to $x\in V$.
Let $L$, $F$, and $R$ be the lowering, flat, and raising matrices, respectively; cf.~\eqref{quantum decomposition}.
The \emph{quantum decomposition}\footnote{The quantum decomposition for the complete graph $K_2$ is sometimes referred to as a \emph{quantum coin-tossing}.} of $A$ is the expression $A=L+F+R$.
The primary $\TT$-module, together with $L$ and $R$, naturally has the structure of a one-mode interacting Fock space,\footnote{In this context, $L$ and $R$ are called the \emph{annihilation} and the \emph{creation operators}, respectively.} and they took the limit of the coefficients of the three-term recurrence relation of the associated orthogonal polynomials (which are certain normalizations of the $v_i$ from \eqref{distancepolynomials}) to get the \emph{quantum central limit theorem}.
See \cite{HO2007B,HO2008TAMS} for more details.
For the above four families of distance-regular graphs, these orthogonal polynomials belong to the Askey scheme by virtue of Leonard's theorem (see also the comments after \eqref{TTR}), and their results agree with the limit relations of the polynomials in the Askey scheme as described in \cite{KS1998R,KLS2010B}.
We note that they also considered some other states as well; see \cite{Hora2000PTRF,HO2007B}.
The case of the Odd graphs was discussed in detail by Igarashi and Obata \cite{IO2006P}.
Associated to the Odd graphs are the Bannai/Ito polynomials (which are a $q\rightarrow -1$ limit of the most general $q$-Racah polynomials), and the generalized Hermite polynomials arise as the orthogonal polynomials corresponding to the limit distribution.
See also \cite{GVZ2014SIGMA}.


\subsection{Spin models}\label{sec:spinmodels}

A (symmetric) \emph{spin model} is a nowhere-zero symmetric matrix $W\in M_{v\times v}(\mathbb{C})$ which
satisfies certain `invariance equations', and was introduced by Jones \cite{Jones1989PJM} as a tool for
creating invariants of knots and links. Nomura \cite{Nomura1997JAC} showed that every spin model $W$ belongs
to its \emph{Nomura algebra} $\mathcal{N}_W\subset M_{v\times v}(\mathbb{C})$, which is the Bose-Mesner
algebra (over $\mathbb{C}$) of a self-dual association scheme; see also
\cite{Jaeger1996P,JMN1998JAC,CGM2003TAMS}. We say that a distance-regular graph $\G$ with diameter $D$
\emph{supports} a spin model $W$ if its adjacency algebra $\AL$ (over $\mathbb{C}$) satisfies
$W\in\AL\subset\mathcal{N}_W$. In this case, $\AL$ inherits the duality of $\mathcal{N}_W$. In particular,
$\G$ is $Q$-polynomial. Many examples of spin models have been constructed in this situation; cf.~\cite[\S
9]{CN1999JCTB}. Write $W=\sum_{i=0}^Dt_iA_i$, where $A_i$ is the distance-$i$ matrix of $\G$ for
$i=0,1,\dots,D$. Curtin and Nomura \cite{CN1999JCTB} showed among other results that if $t_1\ne\pm t_0$ then
the intersection array of $\G$ is described by $t_1/t_0$, $t_0t_2/t_1^2$, and $D$. Curtin \cite{Curtin1999DM}
showed that $\G$ is thin if $t_i\ne\pm t_0$ for $i=1,2,\dots,D$, and Caughman and Wolff \cite{CW2005JAC}
determined the structure of the Terwilliger algebra $\TT$. Moreover, Curtin \cite[Thm.~1.6]{Curtin2007RJ}
showed that, in view of \cite[Thm.~5.3]{CW2005JAC}, every irreducible $\TT$-module affords not just a Leonard
system (cf.~Section \ref{sec: TD systems}) but a \emph{Leonard triple system} \cite{Curtin2007LAA}. See
\cite{Huang2012pre,Brown2013pre,TZ2013pre} and the references therein for related results.
Nomura \cite{Nomura1995JCTB,Nomura1996P} and Curtin and Nomura \cite{CN2004JAC} studied the homogeneity of
$\G$. It is known (cf.~\cite[\S 4.4]{Jaeger1996P}) that the diagonal matrix $T$ of size $D+1$ defined by
$T_{ii}=t_i$ ($i=0,1,\dots,D$) satisfies the \emph{modular invariance property}, i.e., $(PT)^3$ is a scalar
matrix, where $P=Q$ is the eigenmatrix of $\G$. (Recall $P^2=vI$.) Chihara and Stanton \cite{CS1995GC} showed
that $\G$ has at most $12$ (diagonal) solutions $T$ to $(PT)^3=I$ in general, and classified the solutions
when $\G$ is a forms graph or a Hamming graph. See also \cite{Nomura2002KJM}.

\subsection{Cometric association schemes}\label{sec:cometricschemes}

Distance-regular graphs form the class of metric (or $P$-polynomial) association schemes. Cometric (or
$Q$-polynomial) association schemes are the `dual version' of distance-regular graphs, but the systematic
study of cometric (but not necessarily metric) association schemes has begun rather recently. One of the
pioneers in this area is Suzuki \cite{Suzuki1998JACa,Suzuki1998JACb}, who studied imprimitive cometric
association schemes and association schemes with multiple $Q$-polynomial orderings, using a method of Dickie
\cite{Dickie1995D} based on matrix identities; cf.~\eqref{3-tensor}. In particular, he showed that an
imprimitive cometric association scheme with $D\geq 7$ and with first multiplicity $m_1>2$ is $Q$-bipartite
and/or $Q$-antipodal; cf.~Theorem \ref{prop:imprimitive}. In his classification, there remained two cases of
open parameter sets with $D\in\{4,6\}$. These were recently ruled out by Cerzo and Suzuki \cite{CS2009EJC}
for $D=4$, and by Tanaka and Tanaka \cite{TT2011EJC} for $D=6$. Similarly, there was an open case in the
classification of association schemes with multiple $Q$-polynomial orderings, which was recently ruled out by
Ma and Wang \cite{MaWang5Q}. Thus, the situation is dual to that of Section \ref{sec:2Porder}. Van Dam,
Martin, and Muzychuk \cite{DMM2010pre} studied cometric $Q$-antipodal association schemes and showed that
these are uniform and related to linked systems. Three-class cometric $Q$-antipodal association schemes for
example are equivalent to linked systems of symmetric designs.
LeCompte, Martin, and Owens \cite{LMO2010} showed that  four-class cometric $Q$-antipodal and
$Q$-bipartite association schemes are equivalent to real mutually unbiased bases.
See also \cite{Suda2009pre}. See \cite{MMW07,DMM2010pre} for a comprehensive study on imprimitive cometric
association schemes. Martin and Williford \cite{MW2009EJC} proved the dual of the Bannai-Ito conjecture
discussed in Section \ref{sec:BIconjecture}: There are finitely many cometric association schemes with fixed
first multiplicity at least three. Kurihara \cite{Kur2011T} obtained a dual version of the spectral excess
theorem discussed in Section \ref{sec:spectralexcess}. See also \cite{KN2012JCTA,Nozaki2013pre}.

Concerning constructions of cometric (but not metric) association schemes, the main sources are block
designs, spherical designs, real mutually unbiased bases, and hemisystems and other strongly regular
decompositions of a strongly regular graph; see a survey by Bannai and Bannai \cite{BB2009EJC}, and also the
online table by Martin \cite{Martin2010www}. The `bipartite doubles' of the association schemes of the
Hermitian dual polar graphs $^2\A_{2D-1}(\sqrt{q})$ provide an infinite family of cometric but not metric
association schemes with unbounded diameter; cf.~\cite[pp.~313--315]{bi}. The `extended $Q$-bipartite double'
construction was introduced and worked out in detail by Martin, Muzychuk, and Williford \cite{MMW07}.
Penttila and Williford \cite{PenWil2011} constructed the first known infinite family of primitive cometric
association schemes that are not metric. Hollmann and Xiang \cite{HX2006JAC} earlier constructed a family of
$3$-class association schemes that have the same parameters as those found by Penttila and Williford, without
realizing it was cometric. See also \cite{Cossidente2013JAC}.
Recently, Moorhouse and Williford \cite{Williford2014AGT} constructed an infinite family of cometric $Q$-bipartite association schemes as certain `double covers' of the association schemes of the symplectic dual polar graphs $\C_D(q)$ with $q\equiv 1\pmod{4}$.
These association schemes have two $Q$-polynomial orderings, and are not metric; cf.~Section \ref{sec:Qmultipleordering}.
Such a `double cover' has a quadratic splitting field when $q$ is a non-square, and Moorhouse and Williford asked whether or not it is in general the `extended $Q$-bipartite double' of a primitive cometric but not metric association scheme when $q$ is a square.
If this is indeed the case, then this family would provide counterexamples to the conjecture of Bannai and Ito \cite[p.~312]{bi} mentioned at the end of Section \ref{sec:2Qpol}.

We refer the reader to \cite{BB2009EJC,martintanaka,DMM2010pre} for more
information and recent updates on cometric association schemes.
