%\section{Multiplicities}\label{sec:mult}

\subsection{Terwilliger's tree bound}\label{sec:tree_bound}

Terwilliger \cite{TerMult82} showed that if a distance-regular graph $\G$, say with valency $k$, contains an
isometric subgraph that is also a tree, then the multiplicity of each eigenvalue $\theta \neq \pm k$ of $\G$ is at
least the number of leaves (i.e., vertices of valency one) of that tree. This has been generalized by Hiraki and Koolen
\cite[Prop.~3.1]{HK02} to the case where the subgraph is a block graph (i.e., a graph whose $2$-connected components
are complete). Their result is too technical to state here, however, we mention its following consequence.

\begin{prop}{\em (cf.~\cite[Prop.~3.3]{HK02})}
Let $\G$ be a distance-regular graph of order $(s,t)$ and with
head $h$.
Let $\theta \neq k$ be an
eigenvalue with multiplicity $m$, and let $n=\lfloor \frac{h+1}{2} \rfloor$. Then the following hold:
\begin{enumerate}[{\em (i)}]
\item If $h$ is odd and $\theta \neq -t-1$, then
$m \geq (t+1)t^{n-1}s^n$,
\item If $h$ is even and $\theta \neq -t-1$, then
$m \geq (s+1)(st)^n$,
\item If $h$ is odd and $\theta = -t-1$, then
$m \geq 1+(t+1)(s-1)\frac{(st)^{n}-1}{st-1}$,
\item If $h$ is even and $\theta = -t-1$, then
$m \geq \frac{(s-1)(s+1)((st)^{n+1}-1)}{(st-1)s} + \frac{1}{s}$.
\end{enumerate}
\end{prop}

\noindent This generalizes a result of Zhu \cite[Prop.~3.5]{Zhu93} who obtained that $m\geq (t+1)(s-1)$ for $n=1$. It
also generalizes a result of Bannai and Ito \cite{BI87} who showed
that if $a_1 \neq 0$, then $m \geq (k/2)^n$.

C\'{a}mara, Van Dam, Koolen, and Park
\cite{CDKP2013} showed that in a $1$-walk-regular graph with valency $k$
and an eigenvalue $\theta\neq k$ with multiplicity $m$, a clique
can have size at most $m+1$. This result is well-known for
distance-regular graphs. Powers \cite{Pow88} already observed
earlier that for distance-regular graphs equality in this clique
bound cannot occur if $\theta$ is the second eigenvalue, except for
the complete graph. We can generalize this as follows.

\begin{prop}
Let $\G$ be a distance-regular graph with valency $k$. If $\G$
contains a clique with $c$ vertices, and $\theta \neq k$ is an
eigenvalue of $\G$ with multiplicity $m$, then $c \leq m+1$, with
equality only if $\theta=\theta_{\min}$ and $\G$ is complete,
complete multipartite, or bipartite.
\end{prop}

\begin{proof} Consider a clique $C$ with $c$ vertices, the
idempotent $E=UU^{\top}$ and standard sequence $(u_i)_{i=0}^D$
corresponding to $\theta$. Recall from the proof of Biggs' formula
(Theorem
\ref{Biggsformula}) that $E =\sum_{i=0}^D
\nu_i A_i$, where $\nu_i = \nu_0 u_i$ for $i=0, 1,
\ldots, D$. The submatrix of $E$ indexed by the vertices of $C$
equals $\nu_0(I+u_1(J-I))$, which has rank at least $c-1$ (recall
that $u_1=\theta/k \neq 1$). Because the rank of $E$ equals $m$,
the bound $c\leq m+1$ follows. If equality holds then
$1+u_1(c-1)=0$, and hence $c=1-k/\theta$, which implies that $C$ is
a Delsarte clique and $\theta=\theta_{\min}$.

Suppose now that $\G$ is not a complete graph. We aim to show first
that $u_2=1$. Consider a representation associated to $\theta$ (see
Section \ref{sec2:evmult}); for simplicity we normalize it so that
the vectors $\hat{x}$ have length one for all $x \in V$, and the
inner products between these vectors are given by the standard
sequence. Because the rank of the above submatrix of $E$ is $m$, it
follows that the vectors $\hat{z}$, with $z
\in C$ span the row space of $U$. In particular, if we consider a vertex $x$ at distance one from $C$,
then $\hat{x}=\sum_{z \in C}\alpha_z \hat{z}$ for certain
$\alpha_z$. By taking inner products with $\hat{z}$, it follows
that $\alpha_z$ depends only on whether $x$ is adjacent to $z$ or
not. Hence, because $\sum_{z \in C} \hat{z} =0$,
see
\eqref{eq:repdelsarte}, we may assume that $\alpha_z=0$ for $z
\nsim x$. Now let $y$ be a vertex in $C$ that is not adjacent to
$x$. We then obtain that $$1-u_2=\langle
\hat{x},\hat{x} \rangle-\langle
\hat{x},\hat{y} \rangle=\sum_{z \in C}\alpha_z(\langle
 \hat{z},\hat{x} \rangle-\langle
 \hat{z},\hat{y} \rangle)=0,$$ and hence indeed $u_2=1$.

From
\eqref{standard_sequence}, it now follows that $a_1=k+\theta$, and
hence the polynomial $v_2(z)=\frac{1}{c_2}(z^2-a_1z-k)$ from
\eqref{distancepolynomials} satisfies $v_2(\theta)=v_2(k)$,
which implies that $\G_2$ is disconnected. If the diameter equals
two, then $G$ is a complete multipartite graph, so we may now
assume that $D>2$.

Because $\G_2$ is disconnected, it follows that $a_2=0$, for
otherwise $p_{22}^1>0$, which would imply that from every path
between two given vertices in $\G$ one can construct a path in
$\G_2$ between these two vertices. Suppose now that $a_1>0$. Let
$x_0
\sim x_1\sim x_2\sim x_3$ be a shortest path between two vertices $x_0$ and
$x_3$ at distance three, and let $y$ be a common neighbor of $x_1$
and $x_2$. Because $a_2$ is zero, it follows that $y$ is also
adjacent to $x_0$ and $x_3$, and so the latter are not at distance
three, which is a contradiction. Thus $a_1=0$, and because
$a_1=k+\theta$, it follows that $\theta=-k$, and hence $\G$ is
bipartite.
\end{proof}

We remark that the above proof, and hence the result, is also valid
for $2$-walk-regular graphs, just like part of
Godsil's bound in the next section (cf.~Section
\ref{sec:almostdrg}).

\subsection{Godsil's bound}

Godsil \cite{God88} obtained the following lower bound on the
multiplicity of an eigenvalue.
\begin{theorem}\label{thm:godsilbound}{\em (Godsil's bound)}
Let $\G$ be a distance-regular graph with valency $k$ and diameter
$D$, and suppose $\G$ is not a complete
multipartite graph. If $\G$ has an eigenvalue with multiplicity $m
\geq 3$, then $D \leq 3m-4$ and $k \leq
\frac{(m+2)(m-1)}{2}$.
\end{theorem}

\noindent Godsil's diameter bound was improved by Hiraki and Koolen \cite{HK02}.

\begin{prop} {\em \cite[Thm.~1.1,~1.2]{HK02}}
Let $\G$ be a distance-regular graph with diameter $D$. If $\G$ has
an eigenvalue with multiplicity $m \geq 3$, then $D \leq m+6$, unless $h=1$ and $c_2 =1$, in which case $D < m+2 + \log_5 m$.
\end{prop}

\noindent Note that the Doubled Odd graph with valency $k$ has diameter $2k-1$ and an
eigenvalue $k-1$ with multiplicity $2k-2$, so this result is close to the
best possible. Yet another lower bound is obtained by Juri\v si\' c, Terwilliger, and \v{Z}itnik \cite{JTZ2010EJC} (cf.~Section \ref{sec: Hadamard products}):
\begin{prop}\label{light tail bound}
Let $\G$ be a distance-regular graph with valency $k$. If $\theta \neq \pm k$ is an eigenvalue of $\G$
with multiplicity $m$, then
\begin{equation*}
	m \geq k - \frac{a_1 k (\theta +1)^2}{(k+ \theta)^2 + a_1 (\theta^2 - k)}.
\end{equation*}
\end{prop}
%
Koolen, Kim, and Park \cite{KoKiPapre} refined the above valency bound of Godsil. By using the theory developed by
Juri\v si\' c et al.~\cite{JTZ2010EJC}, they were able to show that for $k \geq 3$ and $m \geq 3$, the only possible
distance-regular graphs with diameter at least 3 and $k =\frac{(m+2)(m-1)}{2}$ are Taylor graphs with intersection
array $\{ (2\alpha +1)^2(2\alpha^2 + 2 \alpha -1), 2\alpha^3(2\alpha +3), 1; 1, 2\alpha^3(2\alpha +3), (2\alpha
+1)^2(2\alpha^2 + 2 \alpha -1)\}$, with $m = 4\alpha^2 + 4 \alpha -1$, where $\alpha$ is an integer not equal to $0$
and $-1$ or $\alpha = \frac{-1\pm \sqrt{5}}{2}$ ($m=3$).

Terwilliger (cf.~\cite[Thm.~4.4.4]{bcn}) showed that if a distance-regular graph with valency $k$ has an eigenvalue
$\theta \neq k$ with multiplicity $m <k$, then $\theta$ is either the second largest or the smallest eigenvalue.
Moreover, in this case $-1 - \frac{b_1}{\theta+1}$ is an algebraic integer as it is an eigenvalue of a local graph.
Also, if $m \leq (k-1)/2$, then $\theta$ is an integer such that $\theta +1$ divides $b_1$. Terwilliger's result was
slightly improved by Godsil and Hensel \cite{GodsilHensel92} for antipodal distance-regular graphs, and by Godsil and
Koolen \cite{GoKo95} for the case that $a_D= 0$. As a consequence, Godsil and Koolen showed that a distance-regular
graph with intersection array $\{\mu(2\mu+1), (\mu-1)(2\mu +1), \mu^2, \mu; 1, \mu, \mu(\mu-1), \mu(2\mu +1) \}$ with
$\mu \geq 2$ does not exist.


\subsection{The distance-regular graphs with a small multiplicity}
Let $\G$ be a distance-regular graph with valency $k$. The eigenvalues $k$, and $-k$ in case $\G$ is bipartite,
are the only eigenvalues with multiplicity one. Each eigenvalue of a polygon, besides $\pm k$, has multiplicity two;
and the polygons are the only distance-regular graphs with an eigenvalue having multiplicity two. The five Platonic
solids, i.e., the icosahedron, dodecahedron, cube, octahedron, and tetrahedron are the only distance-regular graphs
with an eigenvalue having multiplicity three. Zhu \cite{Zhu93} (see also \cite{Zhuthesis}) determined the
distance-regular graphs with an eigenvalue having multiplicity four, whereas Martin and Zhu \cite{MarZhupre} (see also
\cite[Ch.~7]{Koothesis}) determined those with an eigenvalue having multiplicity five, six, or seven. Koolen and Martin
\cite{KooMarpre} (see also \cite[Ch.~7]{Koothesis}) determined the distance-regular graphs with an eigenvalue having
multiplicity eight.

\subsection{Integrality of multiplicities}\label{sec:integralmultiplicities}
Biggs' formula (Theorem \ref{Biggsformula}) for the multiplicities of the eigenvalues
and the requirement that these multiplicities are positive integers
is a crucial part of Biggs' definition \cite[Def.~21.5]{biggs} of feasible
intersection arrays of distance-regular graphs.

Godsil and McKay \cite{gmk} generalized Biggs' formula to walk-regular
graphs, thus obtaining feasibility conditions for such graphs. Recall that a
graph is called walk-regular if the number of closed walks of given length from
a vertex to itself is independent of the chosen vertex but depends only on the
length, $\ell$ say; in other words, every power $A^{\ell}$ of the adjacency
matrix has constant diagonal.

Chv\'{a}tal \cite[Thm.~3]{Chvatal} showed that for strongly regular graphs,
Biggs' feasibility condition of integer multiplicities implies the condition
that the number of closed walks of length $p$ is divisible by $p$ for every
prime $p$. The latter condition is essentially a condition on the spectrum
because the number of closed walks of length $p$ equals $\tr A^p =\sum_{i=0}^d
m_i \theta_i^p.$

Here we generalize Chv\'{a}tal's result as follows.

\begin{prop} Let $\{\theta_0^{m_0},\theta_1^{m_1},\dots,\theta_d^{m_d}\}$ be the
multiset of roots of a monic polynomial with coefficients in ${\mathbb Z}$. If
$\sum_{i=0}^d m_i\theta_i=0$, then every prime $p$ divides $\sum_{i=0}^d m_i\theta_i^p$.
\end{prop}

\begin{proof}
By grouping algebraic conjugates, say $\Theta_i$ is the set of algebraic
conjugates of $\theta_i$, and observing that $\sum_{\theta \in \Theta_i}
\theta^p \equiv (\sum_{\theta \in \Theta_i} \theta)^p \equiv (\sum_{\theta \in
\Theta_i} \theta) \mod p$ (the latter equality is by Fermat), it follows that
$\sum_{i=0}^d m_i \theta_i^p \equiv \sum_{i=0}^d m_i \theta_i \equiv 0 \mod p$.
\end{proof}

For a distance-regular graph, both the eigenvalues with multiplicities and the numbers of walks of length $p$
follow from its intersection array. When one wants to generate putative intersection arrays for
distance-regular graphs, testing the integrality of multiplicities is an important but computationally
expensive part. The proposition indicates that testing integrality of the multiplicities is stronger than
testing that the number of closed walks of length $p$ is divisible by $p$. Note that given the intersection
array it is relatively easy to compute the number of closed walks of given length recursively by using the
distance polynomials $v_i$ in \eqref{distancepolynomials} and their sum, the Hoffman polynomial (hence it is
not necessary to first compute the eigenvalues and multiplicities); cf.~\cite[Thm.~2]{Chvatal}. Brouwer,
Cohen, and Neumaier \cite[p.~134]{bcn} give the intersection array $\{26,25,5,1;1,5,25,26\}$ for a
distance-regular graph for which some of the eigenvalues have irrational multiplicities. We found that the
number of closed walks of length $7$ is not divisible by $7$ in this example. An example of an intersection
array that survives the tests on the number of walks is $\{18,15;1,2\}$; this corresponds to a strongly
regular graph with parameters $(v,k,\lambda,\mu)=(154,18,2,2)$. Indeed, the number of closed walks of length
$\ell$ equals $18(18^{\ell-1}-4^{\ell-1})$ for odd $\ell$, and $18^2+153\cdot4^2$ for $\ell=2$. However, the
eigenvalues $4$ and $-4$ have multiplicities $74.25$ and $78.75$, respectively.
