
%\section{More combinatorial properties}\label{sec:morecombinatorial}

\subsection{Distance-regular graphs with a relatively small number of vertices} The Taylor graphs and Hadamard graphs
form infinite families of graphs that have a relative small number of vertices compared to the valency $k$. Indeed,
Taylor graphs have $2k+2$ vertices and Hadamard graphs have $4k$ vertices. The following result, obtained by Koolen and
Park \cite{KoPapre12}, shows that these two families are exceptions.
\begin{theorem}
Let $\alpha > 2$. Then there are finitely many distance-regular graphs with $v$ vertices, valency $k$, diameter $D \geq
3$ satisfying $v \leq \alpha k$, besides imprimitive distance-regular graphs with diameter $3$ and antipodal bipartite
distance-regular graphs with diameter $4$.
\end{theorem}
%
\noindent As a consequence, they also obtained the following.
%
\begin{theorem}
Let $0 < \epsilon < 1$. Then there are finitely many distance-regular graphs with valency $k \geq 3$, diameter $D \geq
3$ satisfying $c_2 \geq \epsilon k$, besides imprimitive distance-regular graphs with diameter $3$ and antipodal
bipartite distance-regular graphs with diameter $4$.
\end{theorem}
%
\noindent For $k \leq 1/ \epsilon$, this result follows from the Bannai-Ito conjecture (see Section
\ref{sec:proofBIconjecture}). If $k
> 1/\epsilon,$ then $c_2 \geq 2$ and one can use the Ivanov bound (Theorem \ref{ivanovbound}) to bound the diameter, and hence  one
can bound the number of vertices by a constant times $k$.

In the case that $\G$ contains a quadrangle, Koolen and Park \cite{KoPapre} obtained the following bound on $c_2$ in
terms of the valency and the diameter.

\begin{prop}\label{c2}
Let $\Gamma$ be a distance-regular graph with valency $k \geq 3$ and diameter
$D \geq 4$. If $\Gamma$ contains an induced quadrangle, then $c_2 \leq
\frac{2}{D} k$ with equality if and only if $D \geq 5$ and $\Gamma$ is a
$D$-cube or $D=4$ and $\Gamma$ is a Hadamard graph.
\end{prop}

\noindent
The assumption of having induced quadrangles is necessary as the
Foster graph and the Biggs-Smith graph have $k =3$, $c_2 =1$ and $D \geq 7$.
We wonder whether the assumption can be removed for $k$ large
enough.

Note also that diameter three is exceptional, because the complete bipartite
graph $K_{k+1,k+1}$ minus a perfect matching has valency $k$ and $c_2=k-1$.
Koolen and Park \cite{KoPa11} showed that if a distance-regular graph has
diameter three, then $c_2 \leq k/2$ or it is bipartite or a Taylor graph. They
also showed that if a distance-regular graph with diameter at least three and
valency $k$ has $a_1 \geq \frac{k-2}{2}$, then it is a Taylor graph, a line
graph, the Johnson graph $J(7,3)$, or the halved 7-cube.

For $4 \leq D \leq 6$, Koolen and Park \cite{KoPapre} strengthened Proposition
\ref{c2} as follows:

\begin{prop}
Let $\Gamma$ be a distance-regular graph with valency $k \geq 3$ and
diameter $D$.
Then the following hold:
\begin{enumerate}[{\em (i)}]
\item If $D \geq 6$ and $c_2 \geq 2$, then $c_2 \leq k/3$,
\item If  $D =4$ and $c_2 > k/3$ then  $\Gamma$ is a Hadamard graph (and hence
$c_2 = k/2$),
\item If $D =5$ and $c_2 > k/3$, then $\Gamma$ is the $5$-cube,
\item If $D=6$ and $c_2 > 2k/7$, then $\Gamma$ is the $6$-cube or the
    generalized dodecagon of order $(1,2)$.
\end{enumerate}
\end{prop}




\subsection{Distance-regular graphs with multiple \texorpdfstring{$P$-polynomial}{P-polynomial} orderings}\label{sec:2Porder}
In the following, we assume that both the diameter and valency of $\G$ are at least three, and we follow `BCN' \cite[\S
4.2.D]{bcn}. If a distance-regular graph $\Gamma$ has a second $P$-polynomial ordering then the corresponding
distance-regular graph $\Delta$ with the second ordering is either the distance-2 graph $\G_2$, the
distance-$(D-1)$ graph $\G_{D-1}$, or the distance-$D$ graph $\G_D$.

The first case ($\Delta=\G_{2}$) is only possible if $\Delta$ is a Taylor graph (with $c_2 < k-1$) or a generalized odd
graph.

The second case ($\Delta=\G_{D-1}$) occurs if and only if $\Gamma$ is an antipodal 2-cover with diameter $D$ such that
$a_i = 0$ for all $i < (D-1)/2$. In this case, the folded graph is either bipartite or a generalized odd graph. If the
folded graph is bipartite, then $\Gamma$ is bipartite and the diameter $D$ is even. For diameter 4 only the Hadamard
graphs occur; for diameter 6 only the 6-cube. For larger diameter only the $D$-cubes are known. If the folded graph is
a generalized odd graph with diameter 3 then $\Gamma$ is a Taylor graph. For larger diameter and bipartite $\Gamma$
only the Doubled Odd graphs and the cubes are known. If $\Gamma$ is not bipartite, then only three graphs are known:
the Wells graph ($D=4)$, the dodecahedron $(D=5)$, and the coset graph of the truncated even subcode of the binary
Golay code (see \cite[p.~365]{bcn}).

For the last case ($\Delta=\G_{D}$) either $\Delta$ is a generalized odd graph or $a_D=0$. In the latter case
($a_D=0$), it holds that $p^D_{2D} \neq 0$ and if $D =4$ then $p^4_{34} = 0$; moreover, Suzuki \cite{Suzuki1996JCTA}
showed that $D \leq 4$.

\subsection{Characterizing antipodality and the height}\label{sec:charantipodal}

Let $\Gamma$ be a distance-regular graph with {valency at least $3$. It is well-known that $\Gamma$ is an antipodal
$2$-cover if and only if $b_{D-i} = c_i$ for all $i =1,2,\dots,D$ \cite{G74}. Araya and Hiraki \cite{AH98} improved
this by showing that $\Gamma$ is an antipodal 2-cover if and only if $b_{D-i} = c_i$ for $i =1,2,\dots,\lceil D/2
\rceil$. This also improved earlier work of Araya, Hiraki, and Juri\v si\'c \cite{AHJ96}, who showed that a
distance-regular graph is an antipodal 2-cover if there is a $j$ with $b_j =1$ and $D \geq 2j$. Araya, Hiraki, and
Juri\v si\'c \cite{AHJ97} also showed that if $b_2=1$, then $\G$ is an antipodal cover, in particular it is either an
antipodal cover of a complete graph ($D=3$), an antipodal $2$-cover of a strongly regular graph with $\lambda =0$ and
$\mu = 2$ ($D=4$), or the dodecahedron ($D=5$). This solved one of the problems in `BCN' \cite[Prob.~(i),~p.~182]{bcn}.

Suzuki \cite{S294} showed that if $k_i = k_j$ for some $i < j$ with $i+j \leq D$, then either $k_D=1$ or $k_i = k_{i+1}
= \cdots = k_j$, thus solving another problem in `BCN' \cite[Prob.~(ii),~p.~168]{bcn}. The only known distance-regular
graphs with $k_i = k_j$ for some $i < j$ with $i+j \leq D$ and $k_D \geq 2$ are the odd polygons, but it is unknown
whether any others could exist. Hiraki, Suzuki, and Wajima \cite{HSW95} showed that if $k_2=k_j$ for $2+j\leq D$ and
$2<j$, then $\Gamma$ is indeed a polygon $(k=2)$ or an antipodal 2-cover ($k_D=1$). In order to show this result, the
height of a distance-regular graph was used; a notion that we will introduce next.

The {\em height} $\text{ht}(\Gamma)$ of a distance-regular graph $\Gamma$ with diameter $D$ is defined as the maximal
$i$ such that the intersection number $p^D_{Di}$ is nonzero. Note that $\Gamma$ is an antipodal 2-cover if and only if
$\text{ht}(\Gamma) = 0$.  The case $\text{ht}(\Gamma) = 1$ occurs exactly when the distance-$D$ graph $\G_D$ is a
generalized odd graph; see \cite[Prop.~4.2.10]{bcn}. Nakano \cite{N01} strengthened some of the results in \cite{HSW95}
by showing that if $k_i=k_j$ for some $i$ and $j$ such that $i<j\leq D-i$ and $\text{ht}(\Gamma)$ is even and at most
$2(D-2i)$, then this height must be zero, that is, $k_D = 1$.

Suzuki \cite{S594} asked whether $\Gamma$ can be characterized by its induced
subgraphs on $\Gamma_D(x)$, for $x \in V$. Some results in this direction were
obtained by Hiraki \cite{Hi299}, who showed that if every induced subgraph on
$\Gamma_{D,\text{ht}(\Gamma)}(x,y)$ is a clique whenever $d(x,y) = D$, then
$\text{ht}(\Gamma) = D$, $D-1$, or $1$.  He also showed that if
$p^D_{D,\text{ht}(\Gamma)} = 1$, then $\text{ht}(\Gamma)$ equals $D$, $1$, or
$0$. Also Tomiyama \cite{To96, To98} gave some results in this direction, that
is, for the case $\text{ht}(\Gamma) = 2$.

\subsection{Bounds on \texorpdfstring{$k_D$}{kD} for primitive distance-regular
graphs}\label{sec:boundskD}

Let $\G$ be distance-regular with diameter $D$ and valency $k$. Brouwer et al.~\cite[Prop.~5.6.1]{bcn} showed that if
$\Gamma$ is not antipodal, then $k \leq k_D(k_D-1)$. Suzuki \cite{Su91, S594} showed that in this case also the
diameter is bounded by a function of $k_D$. This now also follows from the above statement and the validity of the
Bannai-Ito conjecture (see Section \ref{sec:BIconjecture}).

Park \cite{Papre12} showed that if $\G$ has valency and diameter at least $3$ and satisfies $k_{D-1} + k_{D} \leq 2k$,
then $\G$ is an antipodal $2$-cover, $\G$ is bipartite with $D=3$, $\G$ is the Johnson graph $J(7,3)$, or $\G$ is the
halved 7-cube. In the case $D=3$, there are infinitely many bipartite non-antipodal distance-regular graphs with $k_2 +
k_3 \leq 2k$, for example the incidence graphs of the complements of projective planes of order at least $3$. This
result also confirms a conjecture by Bendito, Carmona, Encinas, and Mitjana \cite{BCEM12} that states that no primitive
distance-regular graph with diameter three has the so-called $M$-property.




\subsection{Terwilliger graphs and existence of quadrangles}\label{sec:existencequadranglesT}

Recall that a distance-regular graph without induced quadrangles is called a {\em Terwilliger graph}. In this section
we collect some sufficient conditions for a distance-regular graph with $c_2 \ge 2$ to contain induced quadrangles.
Note that by Proposition \ref{thm:terwilintersectionnos}, the existence of a quadrangle implies that $c_i - b_i  \geq
c_{i-1} - b_{i-1} + a_1 + 2$. On the other hand, it is shown in the proof of \cite[Thm.~5.4.1]{bcn} that a
distance-regular graph with $c_3 < 2c_2$ and $c_2 \geq 2$ has an induced quadrangle. The following are some more such
combinatorial conditions.
\begin{prop}
Let $\Gamma$ be an amply regular  Terwilliger graph
with diameter $D \geq 2$ and with parameters $(v,k, \lambda, \mu)$ such that $\mu \geq 2$.
%with parameters $(v,k, \lambda, \mu)$ such that $\mu \geq 2$ and with diameter $D \geq 2$.
\begin{enumerate}[{\em (i)}]
\item If  $k \leq (6 + \frac{8}{57} )(\lambda + 1)$, then $\Gamma$ is the icosahedron, the Doro graph (see \cite[\S
    12.1]{bcn}), or the Conway-Smith graph (see \cite[\S 13.2]{bcn}) \cite[Prop.~6]{KoPa11},
\item If $k < 50(\mu-1)$, then $\Gamma$ is the icosahedron, the Doro graph, or the Conway-Smith graph
    \cite[Cor.~1.16.6(ii)]{bcn},
\item If $24\mu > 10(\lambda +1)$, then $\Gamma$ is the icosahedron, the
    Doro graph, or the Conway-Smith graph \cite{kooterwpre}.
\end{enumerate}
\end{prop}

\noindent In \cite[Thm.~4.4.11]{bcn}, the distance-regular graphs with second largest eigenvalue $b_1 -1$ are
classified. For Terwilliger graphs we can go a little further.

\begin{prop} {\em \cite{kooterwpre}}
Let $\Gamma$ be a distance-regular Terwilliger graph with diameter $D \geq 3$
and distinct eigenvalues $\theta_0 > \theta_1 > \cdots > \theta_D$. Then
$\theta_1 \leq b_1/2 -1$ and $\theta_D \geq -b_1/3 -1$, unless $\Gamma$ is the
icosahedron, the Doro graph, or the Conway-Smith graph.
\end{prop}

\noindent Note that if we would know the Terwilliger distance-regular graphs that are locally Hoffman-Singleton, then
we could improve the above results. Note that there are two feasible intersection arrays known which could be locally
Hoffman-Singleton: $\!\{ 50, 42, 9; 1, 2, 42\}$ and $\{ 50, 42, 1; 1, 2, 50\}$, see \cite[p.~36]{bcn}. Gavrilyuk and
Makhnev \cite{GavMak2011} have worked on the classification of these graphs.






\subsection{Connectivity and the second eigenvalue}

\subsubsection{Connectivity and matchings}
Brouwer and Koolen \cite{BrKo09} showed that a (non-complete) distance-regular
graph $\G$ with valency $k>2$ is  $k$-connected and that the only way to
disconnect $\G$ by removing $k$ vertices is to remove the neighborhood of some
vertex. This implies that also the edge-connectivity of $\G$ equals its
valency, and consequently, that every distance-regular graph on an even number
of vertices has a perfect matching. This had been derived before by Brouwer and
Haemers \cite{BH05}, who also showed that the only way to disconnect $\G$ by
removing $k$ edges is to remove the edges through some vertex.

It was noted by Beezer and Farrell \cite{BeFa00} that in general, the number of
perfect matchings does not follow from the intersection array. They showed that
the numbers of matchings consisting of $i$ edges are determined by the
intersection array for $i=1,2,\ldots, 5$; however the Hamming graph $H(2,4)$
and the Shrikhande graph (which have the same intersection array) have
different numbers of matchings with $i$ edges for every $i>5$.

\subsubsection{The second largest eigenvalue}\label{sec:DIAMETEReigenvalues}
Koolen, Park, and Yu \cite{KoPaYu11} showed that for given $\alpha >1$, there
are only finitely many distance-regular graphs with $k \geq 3$ and $D \geq 3$
whose second largest eigenvalue $\theta_1$ satisfies $\alpha \geq \theta_1 >1$.
Note that the (infinite family of) regular complete bipartite graphs minus a
perfect matching are the only distance-regular graphs with $D \geq 3$ and
$\theta_1=1$, and there are no distance-regular graphs with $D \geq 3$ and
$\theta_1<1$. The distance-regular graphs with $D \geq 3$ and $\theta_1 \leq 2$
were also classified.

The distance-regular graphs with $D \geq 3$ and $a_1 \geq 2$ such that all
local graphs have second largest eigenvalue at most one have been classified by
Koolen and Yu \cite{KoYu11}. One may wonder whether, given $\alpha \geq 1$,
there are only finitely many distance-regular graphs with $D \geq 3$ and $a_1 >
\alpha$ such that each local graph has second largest eigenvalue at most
$\alpha$. The condition $a_1 > \alpha$ ensures that the local graphs are
connected, and thus excludes infinite families such as the Hamming graphs $H(D,
\alpha + 2)$.



\subsubsection{The standard sequence}\label{sec:standardsequence}

Let $\Gamma$ be a distance-regular graph with diameter $D$ and let
$$L(i)= \left[
\begin{tabular}{lllllll}
$0$ & $b_0$\\
$c_1$ & $a_1$ & $b_1$\\
  & $c_2$ & $a_2$ & $b_2$\\
  &       & . & . & .\\
  &       &       & . & . & .\\
  &       &       &   & $c_{i-1}$ & $a_{i-1}$ & $b_{i-1}$\\
  &       &       &   &\makebox{\hspace{.324cm}}   & $c_i$  & $a_i$
\end{tabular}
\right],$$


$$M(i)= \left[
\begin{tabular}{llllll}
$a_i$ & $b_i$\\
$c_{i+1}$ & $a_{i+1}$ & $b_{i+1}$\\
  & . & . & .\\
  &       & . & . & .\\
  &       &   & $c_{D-1}$ & $a_{D-1}$ & $b_{D-1}$\\
  &       &   &\makebox{\hspace{.324cm}}   & $c_D$  & $a_D$
\end{tabular}
\right].$$
%
Cioab\u{a} and Koolen \cite{CioaKopre} studied the eigenvalues
of these matrices in order to answer a question by Brouwer. Note that the
eigenvalues of $L= L(D)=M(0)$ are the $D+1$ distinct eigenvalues $\theta_0>
\theta_1 > \cdots
>\theta_D$ of $\Gamma$. Let $\rho_i$ be the largest eigenvalue of $L(i)$, let
$\sigma_i$ be the largest eigenvalue of $M(i)$, and let $u_0=1, u_1, \ldots,
u_D$ be the standard sequence of the second largest eigenvalue $\theta_1$ of
$\G$. By the theory of orthogonal polynomials, it follows that this sequence
has one sign change. Also, for $i= 2,3, \ldots, D-1$ and $\varepsilon \in \{+1,
-1\}$, if $\varepsilon u_i
>0$, then $\varepsilon \rho_{i-1} < \varepsilon \theta_1 < \varepsilon
\sigma_{i+1}$. If $u_i = 0$, then $\theta_1 = \rho_{i-1} = \sigma_{i+1}$.

For $D=3$, this means that $\theta_1$ lies between $a_3$ and
$\frac{a_1 + \sqrt{a_1^2 + 4k}}{2}$, and if two of these three numbers are
equal, then they are all equal. The latter case defines the class of Shilla
graphs introduced by Koolen and Park \cite{KoPa10}.

Cioab\u{a} and Koolen \cite{CioaKopre} used the above to derive
that the induced subgraph $\Xi(j)$ on $\Gamma_j(x) \cup \Gamma_{j+1}(x) \cup
\ldots \cup \Gamma_D(x)$ is connected if $j \leq D/2$, and that $\Xi(D/2+1)$ is
not connected if and only if  $\Gamma$ is an antipodal $r$-cover with $r \geq
3$. This answers a question by Brouwer. It is not clear when $\Xi((D+1)/2)$ is
disconnected.

A final remark on the standard sequence of the second
eigenvalue is that Park, Koolen, and Markowsky \cite{PaKoMar} showed that
$u_{j}> 0$ if $j < D/2$ and $u_j \geq 0$ if $j = D/2$. Moreover, they showed
that $u_{D/2} = 0$ if and only if $\Gamma$ is an antipodal cover.

We remark that if $\theta_1 < \alpha k$, 
then $c_t + a_t < \alpha k$ and hence $b_t = k- (a_t + c_t) > (1-\alpha)k$ for $2t+2 \leq D$.  This implies
that 
$k_t > k \frac{(1-\alpha)^{t-1}}{\alpha^{t-1}}$ for $2t + 2 \leq D$. So if $\theta_1 < k/2$, then $k_t > k_{t-1}$,
which gives a partial answer to a problem in `BCN' \cite[p.~189]{bcn}. If $\theta_1 < k/3$, then we can improve Pyber's
bound of Section \ref{sec:Pyber}.


\subsection{The \texorpdfstring{distance-$D$}{distance-D} graph}\label{sec:distance-D graph}

The spectral excess theorem (cf.~Theorem \ref{spectral excess theorem}) states that a connected regular graph with $d+1$ distinct eigenvalues is distance-regular precisely when the distance-$d$ graph is regular with the `right' valency determined by the spectrum of the graph.
As mentioned in Section \ref{sec:spectralexcess}, Fiol \cite{F00} specialized this theorem to strongly distance-regular graphs.
Fiol \cite[Conj.~3.6]{Fiol2001CPC} also conjectured that a distance-regular graph with diameter at least $4$ is strongly distance-regular if and only if it is antipodal.\footnote{We note that a distance-regular graph with diameter $3$ is strongly distance-regular precisely when it has $-1$ as an eigenvalue; cf.~\cite[Prop.~4.2.17]{bcn}. Some non-antipodal examples are the Odd graph $O_4$ and the Sylvester graph.}

Fiol \cite{Fiol2001CPC} showed that a distance-regular graph with diameter $D$ and distinct eigenvalues $k=\theta_0>\theta_1>\dots>\theta_D$ is strongly distance-regular if and only if, for $i=1,2,\dots,D$, the multiplicity $m_i$ of $\theta_i$ is expressed as a certain rational function in $\theta_0,\theta_1,\dots,\theta_D$, and the number of vertices, $v$.
Brouwer and Fiol \cite{BF2014pre} among other results strengthened this result for the case $D=4$ as follows:

\begin{prop}
Let $\G$ be a distance-regular graph with diameter $4$.
Then the following are equivalent:
\begin{enumerate}[{\em (i)}]
\item $\G$ is strongly distance-regular, i.e., the distance-$4$ graph $\G_4$ is strongly regular,
\item $b_3 = a_4 +1$ and $b_1 = b_3c_3$,
\item $(\theta_1 +1)(\theta_3 +1) = (\theta_2 +1)(\theta_4 +1) = -b_1$.
\end{enumerate}
\end{prop}

\noindent
See also \cite{Fiol2014pre}.
In \cite{BF2014pre}, Brouwer and Fiol in fact studied the more general situation where the distance-$D$ graph has at most $D$ distinct eigenvalues; or equivalently, where the distance-$D$ matrix $A_D$ generates a proper subalgebra of the adjacency algebra $\AL$.
They showed for example that a distance-regular graph with diameter $D$ belongs to this class provided that $D$ is odd and the distance $1$-or-$2$ graph is distance-regular (e.g., the Odd graph $O_{D+1}$, the folded $(2D+1)$-cube, and the dual polar graphs $\B_D(q)$ and $\C_D(q)$).

