%\section{Growth of intersection numbers and diameter bounds}\label{sec:diameterbounds}

In this section we will look at the growth of intersection numbers and its
consequences for bounds on the diameter.

\subsection{The Ivanov bound}
Ivanov \cite{Iv83} obtained the first general diameter bound for
distance-regular graphs.

\begin{theorem} \label{ivanovbound}{\em (The Ivanov bound)}\label{Ivanov}
Let $\G$ be a distance-regular graph with diameter $D \geq 2$, head $h$, and
valency $k$.
Let $2 \leq i \leq i+j \leq D-1$. If $(c_{i-1}, a_{i-1}, b_{i-1}) \neq (c_i, a_i, b_i) = (c_{i+j}, a_{i+j} , b_{i+j})$, then $j \leq i$.
In particular, $D < 2^{k-1}(h +1).$
\end{theorem}

\noindent Suzuki \cite[p.~67]{Su99} gave a proof of this bound using so-called intersection diagrams. Bang, Hiraki, and
Koolen \cite{BHK06, HK98} improved the Ivanov bound, as we shall discuss below. One of the tools that they used is the
following result of Koolen \cite{Ko192}, \cite[Prop.~2.3]{Koothesis}.

\begin{prop}\label{kooprop}
Let $\G$ be a distance-regular graph with diameter $D$.
\begin{enumerate}[{\em (i)}]
\item If $c_i > c_{i-1}$ for some $i =2, \ldots, D$, then $c_{i-j} +c_j \leq c_i$ for all $j=1,\ldots,i-1$,
\item If $b_i > b_{i+1}$ for some $i =0, \ldots, D-2$, then $b_i \geq b_{i+j} +
c_j$ for all $j=1,\ldots, D-i$.
\end{enumerate}
\end{prop}

\noindent Wajima \cite{Waji94} also obtained Proposition \ref{kooprop} (but with a completely different method), and
Hiraki \cite{Hi07} obtained slight improvements of this result. Another tool by Bang et al.~\cite{BHK06} is the
following.

\begin{prop} \label{BHKprop}
Let $\G$ be a distance-regular graph with valency $k$ and diameter $D$. For
$1 \leq c \leq k$, define $\xi_c = \min\{ i : c_i \geq c\}$ and $\eta_c = |\{ i
: c_i = c\}|$. Then $\eta_c \leq \xi_c -1$.
\end{prop}

\noindent Using a combination of  Propositions \ref{kooprop} and \ref{BHKprop}, Bang et al.~\cite{BHK06} improved the
Ivanov bound as follows (using the notation as introduced in Proposition \ref{BHKprop}):

\begin{prop} Let $\G$ be a distance-regular graph with valency $k$ and diameter $D$.
Then $$D < \frac{1}{2}k^{\alpha}\eta_1 +1,$$ where $\alpha = \inf\{ x >0 :
4^{\frac{1}{x}} - 2^{\frac{1}{x}} \leq 1\} \approx 1.441$.
\end{prop}

\noindent Hiraki \cite{Hi01} showed, using earlier work from \cite{CH99, Hi94, Hi98,
HK02}, that if $h \geq 2$, then $c_{2h+3} \geq 2$ or, in other words, $\eta_1
\leq 2h+2$. This  immediately implies that if $h \geq 2$, then $$D \leq
k^{\alpha}(h+1) +1.$$ For $h =1$, it is conjectured by Hiraki \cite{Hi95} that
there exists a constant $C$ such that  $\eta_1 \leq C$. Chen, Hiraki, and Koolen \cite{CHK98}
showed that if $a_1 \neq 2$ and $a_1 \leq 100$, then $c_4 \geq 2$.

In the next sections we present better diameter bounds for certain subclasses
of distance-regular graphs.

\subsection{Distance-regular graphs of order \texorpdfstring{$(s,t)$}{(s,t)}}

The following result was first shown by Terwilliger
\cite{Terwscak} for distance-regular graphs with
$a_1 = 0$ or $c_2 \geq 2$. Later it was generalized by Faradjev, Ivanov, and Ivanov \cite{FII} to
distance-regular graphs with $a_1> 0$. We present their bound for the case that $\G$ is locally a
disjoint union of cliques and $c_{h+1} \geq 2$ holds. In the next section we will
also present the bound of Terwilliger for the case $c_2 \geq 2$.

\begin{prop}{\em (cf.~\cite[Thm.~1.4.3,~Cor.~1.4.4]{Su99})}
Let $\G$ be a distance-regular graph of order $(s,t)$ with
head $h$, valency $k$, and diameter $D \geq 2$. If $c_{h+1} > 1$, then $b_i > b_{i+h}$
and $c_i < c_{i+h}$  for all $i=0,1, \ldots, D-h$, and in particular, $D \leq
t h +1$.
\end{prop}

\noindent For the bipartite case, Koolen \cite{Ko192} and Hiraki
\cite{Hi07} made some improvements. Hiraki \cite{Hi07} showed that
if $\G$ is a bipartite distance-regular graph with head $h \geq
2$ and diameter $D$, then $\G$ is a Doubled Odd graph or $D \leq
\lfloor \frac{k+2}{2} \rfloor h$; see also Section
\ref{sec:otherinfinite}. For the weakly geometric case, Suzuki
obtained the following.

\begin{prop}{\em(cf.~\cite[Prop.~3.1.6]{Su99})}
Let $\G$ be a weakly geometric distance-regular graph of order $(s,t)$ with
head $h$ and diameter $D$. Then $b_i > b_{i+h+1}$ and $c_i < c_{i+h+1}$
for all $i=0,1, \ldots, D-h-1$. In particular, $D \leq t (h+1) +1$.
\end{prop}

\begin{corollary}\label{s>t implies geometric}
Let $\G$ be a distance-regular graph of order $(s,t)$
with head $h$ and diameter $D$. If $s > t$, then $\G$ is
geometric and hence $b_i > b_{i+h+1}$ and $c_i < c_{i+h+1}$ for all
$i=0,1, \ldots, D-h-1$ and in particular,  $D \leq t (h+1) +1$.
\end{corollary}

\begin{corollary} \label{cordiameter} For all integer $t \geq 1$ there exists
a constant $C_t$ such that for all distance-regular graphs $\G$
of order $(s,t)$, the diameter of $\G$ is bounded by $C_t
h(\G)$, where $h(\G)$ is the head of $\G$.
\end{corollary}

\subsection{A bound for distance-regular graphs with \texorpdfstring{$c_2 \geq2$}{c2 ge 2}}\label{sec:boundc2}

Terwilliger \cite{Terwscak} obtained the following bound for
distance-regular graphs with $c_2 \geq 2$.

\begin{prop}{\em (cf.~\cite[Thm.~5.2.5,~Prop.~1.9.1]{bcn})}
Let $\G$ be a distance-regular with diameter $D \geq 2$ and $c_2 \geq
2$. If $c_2 \geq 2(a_1 +1)$, then  $c_i -b_i \geq c_{i-2} - b_{i-2} + 2 \ \ (i
=2, 3, \ldots, D)$, and in particular $D \leq k$. Moreover, if $\max\{a_1, 2\}
\leq c_2$, then $c_i \geq c_{i-1} + 1$ $(i=2,3, \ldots, D)$.
\end{prop}

\noindent Caughman \cite{Cau97} improved this result for bipartite
distance-regular graphs as follows.

\begin{prop}
Let $\G$ be a bipartite distance-regular graph with valency $k$, diameter
$D \geq 3$, and $c_2 \geq 2$. Let $i =1,2,\ldots, D-1$. If $k  > c_i
((c_2-1)(c_2-2)(c_i-c_{i-1}-1)/2 +1)$, then $c_{i+1} \geq c_i(c_2 -1) + 1$.
\end{prop}


\noindent Moreover, Terwilliger \cite{Ter-quad} obtained the following diameter bound.

\begin{prop}\label{thm:terwilintersectionnos} {\em (cf.~\cite[Thm.~5.2.1,~Cor.~5.2.2]{bcn})}
Let $\G$ be a distance-regular graph with diameter $D$. If $\G$
contains an induced quadrangle, then $ c_i - b_i \geq c_{i-1} - b_{i-1} + a_1
+2$ and, in particular, $D \leq \frac{k+c_D}{a_1 +2}$.
\end{prop}

\noindent The distance-regular graphs with diameter $\frac{k+c_D}{a_1 +2}$ and containing a quadrangle have second
largest eigenvalue $b_1-1$ and have been classified: besides the strongly regular graphs with smallest eigenvalue $-2$,
these are the Hamming graphs, Doob graphs, halved cubes, Johnson graphs, locally Petersen graphs, and the Gosset graph,
see \cite[Thm.~5.2.3]{bcn}; also cf.~Section \ref{sec: the case b=1}. Note that if a distance-regular graph contains a
quadrangle then the second largest eigenvalue is at most $b_1-1$.

Neumaier \cite{N04} showed among other results that if there are infinitely
many distance-regular graphs with fixed $a_1, c_2, a_i, c_i$ containing an
induced quadrangle then necessarily $c_{i+1} \geq  1+(c_2-1)c_i$. For dual
polar graphs, equality holds.



\subsection{The Pyber Bound}\label{sec:Pyber}

Using a slightly weaker result than Proposition \ref{kooprop}, Pyber \cite{Py99} showed that $ D \leq 5 \log_2v$ for a
distance-regular graph with $v$ vertices and diameter $D$. This essentially settles a problem in `BCN'
\cite[p.~189]{bcn}. Pyber's bound was improved by Bang, Hiraki, and Koolen \cite{BHK06} to $D < \frac{8}{3} \log_2v$.
