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\journame{Journal of Algebraic Combinatorics}
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\authorrunninghead{A.E. Brouwer \& R.J. Riebeek}
\titlerunninghead{The spectra of Coxeter graphs}

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\title{The spectra of Coxeter graphs}

\authors{A.E. Brouwer \& R.J. Riebeek}
\email{aeb@cwi.nl, remkor@win.tue.nl}

\affil{Department of Mathematics, Eindhoven University of Technology,
P.O. box 513, 5600 MB  Eindhoven, The Netherlands}

\editor{}

\abstract{
We determine the spectra of the finite Coxeter graphs defined by a terminal
node of the Coxeter diagram, and the spectra of their thick equivalents.
}

\keywords{Coxeter Graphs, Graphs of Lie Type, Spectra of Graphs}






F. Buekenhout asked for the spectra of the graphs $F_{4,1}$,
$H_{4,1}$ and $H_{4,4}$ (for notation, see \cite{BCN}, Chapter 10).
In this paper we give the spectra of these and similar graphs.
Some of these spectra are very well known (and given here only
for completeness); we refer to \cite{CDS}, although there may be
older references. In the remaining cases the computations may be
done using Proposition 2.2.2 in \cite{BCN} - however, note that
that proposition as formulated there is valid only when $L_i$
has $d+1$ distinct eigenvalues. A version that is valid in general
(even for non-symmetric association schemes) follows.

\noindent
{\bf Theorem }{\em Let $(X,\{R_0, ... ,R_d\})$ be an association scheme with adjacency matrices $A_0$, ..., $A_d$ (where $A_0 = I$)
and intersection numbers $p_{ij}^k$ (so that $A_i A_j = \sum p_{ij}^k A_k$).
Let $L_i$ be the matrix of order $d+1$ defined by $(L_i)_{kj} = p_{ij}^k$.
Let $A = A_1$ be the adjacency matrix of a symmetric relation $R_1$
in the scheme, and let $L := L_1$. Then the matrices $A$ and $L$
have the same eigenvalues.
If $\lambda$ is an eigenvalue of $L$, then there is a unique real eigenvector
$u$ of $L$ such that $L u = \lambda u$, $u_0 = 1$, $u^{\top} \Delta_n u$
minimal, where $\Delta_n$ is the diagonal matrix with the valencies
$n_j = p_{jj'}^0$ on the diagonal, and the multiplicity of $\lambda$
as an eigenvalue of $A$ equals $v / (u^{\top} \Delta_n u)$.
}

\begin{proof}
The map $A_i \mapsto L_i$ is an isomorphism from the
Bose-Mesner algebra $\fsA = \langle A_i ~|~ 0 \le i \le d \rangle_F$
spanned by the matrices $A_i$ (over some field $F$) onto the algebra
$\langle L_i ~|~ 0 \le i \le d \rangle_F$. Indeed,
it preserves multiplication, and both algebras have the same dimension
since the $L_i$ are independent (because $(L_i)_{k0} = \delta_{i,k}$).
Now let $F = \Rr$.
Both $A$ and $L$ are diagonalizable (since $A$ and $\Delta_n L$ are symmetric);
let $m(X) := \prod_{i=1}^s (X - \theta_i)$ be the minimal polynomial of $A$
(and hence of $L$). Put $m_j(X) := m(X)/(X - \theta_j)$, and
$E_j := m_j(A)/m_j(\theta_j)$. Then the $E_j$ are mutually orthogonal idempotents
in $\fsA$, and $A E_j = \theta_j E_j$.
The multiplicity of $\theta_j$ as an eigenvalue
of $A$ equals $f_j := \rk E_j = \tr E_j$.
Define a $(d+1) \times s$ matrix $Q$ by
$E_j = {1 \over v} \sum_{i=0}^d Q_{ij} A_i$ ($1 \le j \le s$).
Now $L Q e_j = \theta_j Q e_j$ (where $e_j$ is the $j$-th unit vector)
as one sees by comparing the coefficients of the $A_i$ on both sides
of $A E_j = \theta_j E_j$. Furthermore, $(Qe_j)_0 = Q_{0j} = f_j$ follows by
taking traces in the expression for $E_j$.
Finally, $Q^{\top} \Delta_n Q = v \Delta_f$ follows from
$\delta_{ij} f_i = \tr E_i E_j = {1 \over v} \sum Q_{li} Q_{lj} n_l$.
(Note that if $R_l$ and $R_{l'}$ are inverse relations, so that
$A_{l'} = A_l^{\top}$, then $Q_{li} = Q_{l'i}$, since $E_i$ is symmetric.)
Thus, if we take $u = f_j^{-1} (Qe_j)$, then $u_0 = 1$ and
$f_j = v/ (u^{\top} \Delta_n u)$.

Remains to show that this $u$ minimizes $u^{\top} \Delta_n u$.
But if $u$ is arbitrary with $u_0 = 1$, $L u = \theta_j u$, then
the matrix $U := \sum u_i A_i$ satisfies $A U = \theta_j U$ and hence
$U = E_j M$ for some matrix $M$.
Also $u^{\top} \Delta_n u = {1 \over v} \tr U^\top U$ and $u_0 = {1 \over v} \tr U$,
so we have to minimize $\tr U^\top U$ (that is, the sum of the squares of the
elements of $U$) given $\tr U$ (and $U = E_j M$). Diagonalising $E_j$ we see
immediately that $U$ is uniquely determined, and corresponds to the $u$ we had above.
\end{proof}

Thus, it suffices to work with $L_i$ instead of $A_i$.
The required parameters $p_{ij}^k$ for the graphs $E_{6,*}$,
$E_{7,*}$ and $E_{8,*}$ can be found in \cite{BC}.
We give only the results of the computations.

\section{$A_{n,1}$}
The graph $A_{n,1}$ is the complete graph $K_{n+1}$ with spectrum
$n^1 ~ (-1)^n$, where the superscripts denote multiplicities
(cf. \cite{CDS}, p. 72).\\

{\scriptsize

In the thick case $A_{n,1}(q)$ we find the complete graph on $v = \frac{q^{n+1}-1}{q-1}$ vertices with spectrum $(v-1)^1 ~ (-1)^{v-1}$.

}

\section{$B_{n,1}$ and $B_{n,n}$}
The graph $B_{n,1}$ is the complete $n$-partite graph $K_{n \times 2}$
with spectrum $(2n-2)^1 \z 0^n \z (-2)^{n-1}$ (cf. \cite{CDS}, p. 73).

The graph $B_{n,n}$ is the $n$-cube with spectrum
$n^1 \z (n-2)^n ... (n-2j)^{n \choose j} ... (-n)^1$
(cf. \cite{CDS}, p. 75).\\

{\scriptsize

In the thick case $B_{n,1}(q)$ we find the strongly regular polar graph $B_n(q)$ with eigenvalues $q\frac{q^{2n-2}-1}{q-1}$, $-q^{n-1}-1$, $q^{n-1}-1$ and multiplicities (respectively) $1, {1 \over 2}(\frac{q^{2n}-q^2}{q-1} + q^{n-1} +1) , {1 \over 2}(\frac{q^{2n}+q^2}{q-1} - q^{n-1} -1) - {q \over {q-1}}$.

$B_{n,n}(q)$ is the distance-regular dual polar graph $[B_n(q)]$, with eigenvalues\\
$q \qchoose{n-j}{1} - \qchoose{j}{1}$ and multiplicities: $q^j \qchoose{n}{j} \frac{1+ q^{n+1-2j}}{1+q^{n+1-j}} (\prod_{i=1}^{j} \frac{1+q^{n+1-i}}{1+q^{i-1}})$ for $0 \leq j \leq n$. (cf. \cite{BCN}, p. 275)
}

\section{$D_{n,1}$ and $D_{n,n}$}
The graph $D_{n,1}$ is the same as $B_{n,1}$.

The graph $D_{n,n}$ is the halved $n$-cube ${1 \over 2} 2^n$
with eigenvalues $((n-2j)^2 - n)/2$ and multiplicities ${n \choose j}$
$(0 \le j < n/2)$, but multiplicity ${1 \over 2}{n \choose j}$ if $j = n/2$
(cf. \cite{BCN}, p. 264).\\

{\scriptsize

In the thick case $D_{n,1}(q)$ is the strongly regular polar graph $D_n(q)$ with eigenvalues $q\frac{(q^{n-1}-1)(q^{n-2}+1)}{q-1}$, $q^{n-2}-1$, $-q^{n-1}-1$ and multiplicities (respectively) $1, \frac{q^{2n} + 2 q^{n+2} - 2 q^{n+1} - q^2}{q^2 -1}, \frac{q^{2n-1} - 2 q^{n+2} + 3 q^{n+1} - q^{n-1} -q}{q^2-1}$.

$D_{n,n}(q)$ is the halved graph of the dual polar graph $D_n(q)$ with eigenvalues $q^{2j+1} \qchoose{n-2j}{2} - \frac{q^{2j}-1}{q^2-1}$ and multiplicities\\
$\gamma_j q^j \qchoose{n}{j} \frac{1 + q^{n-2j}}{1+q^{n-j}} \prod_{i=1}^{j} \frac{q^{m-i} + 1}{q^i + 1}$, where $0 \leq j \leq {n \over 2}$ and $\gamma_j = 1$ for $2j < m$ and $\gamma_n={1 \over 2}$ if $n$ is even (cf. \cite{BCN}, p. 278).

}

\section{$E_{6,1}$ and $E_{6,2}$}
The graph $E_{6,1}$ is the Schl\"afli graph, strongly regular with
parameters $(v,\<k,\<\lambda,\<\mu) = (27,16,10,8)$ and spectrum
$16^1 \z 4^6 \z (-2)^{20}$ (cf. \cite{BCN}, p. 312).

The graph $E_{6,2}$ is the root system graph of type $E_6$ on 72 vertices.
It has spectrum $20^1 \z 10^6 \z 2^{20} \z (-2)^{30} \z (-4)^{15}$ (cf. \cite{BCN}, p. 313).

{\scriptsize

In the thick case $E_{6,1}(q)$ we find a strongly regular graph
on $v = (q^8 + q^4 + 1)(q^9 - 1)/(q-1)$ vertices, and with valency $q(q^8-1)(q^3+1)/(q-1)$
and spectrum \\

\noindent
eigenvalue: $ q^{11}+q^{10}+q^{9}+2\,q^{8}+2\,q^{7}+2\,q^{6}+2\,q^{5}+2\,q^{4}+q^{3}+q^{2}+q$\\
multiplicity: $ 1 $ \\\\
eigenvalue: $ q^{8}+q^{7}+q^{6}+q^{5}+q^{4}-1$ \\
multiplicity: $q^{11}+q^{8}+q^{7}+q^{5}+q^{4}+q$ \\\\
eigenvalue: $ -q^{3}-1 $ \\
multiplicity: $q^{16}+q^{15}+q^{14}+q^{13}+2\,q^{12}+q^{11}+2\,q^{10}+2\,q^{9}+2\,q^{8}+q^{7}+2\,q^{6}+q^{5}+q^{4}+q^{3}+q^{2}$ \\\\


In the thick case $E_{6,2}(q)$ we find a graph on  
$v = (q^3 + 1)(q^4 + 1)(q^6 + 1)(q^9 -1)/(q-1)$ vertices, and with spectrum\\\\
eigenvalue: $ q^{10}+q^{9}+2\,q^{8}+3\,q^{7}+3\,q^{6}+3\,q^{5}+3\,q^{4}+2\,q^{3}+q^{2}+q $\\
multiplicity: $ 1 $\\\\
eigenvalue: $ q^{8}+2\,q^{7}+2\,q^{6}+3\,q^{5}+2\,q^{4}+q^{3}-1 $\\
multiplicity: $ q^{11}+q^{8}+q^{7}+q^{5}+q^{4}+q $\\\\
eigenvalue: $ q^{7}+q^{6}+q^{5}+q^{4}-q^{2}-1 $\\
multiplicity: $ q^{16}+q^{15}+q^{14}+q^{13}+2\,q^{12}+q^{11}+2\,q^{10}+2\,q^{9}+2\,q^{8}+q^{7}+2\,q^{6}+q^{5}+q^{4}+q^{3}+q^{2}$\\\\
eigenvalue: $ q^{5}-q^{3}-q^{2}-1 $\\
multiplicity: $ {1 \over 2} ( q^{21} + q^{20} + 2\,q^{19} + 3\,q^{18} +
3\,q^{17} + 3\,q^{16} + 5\,q^{15} + 4\,q^{14} + 5\,q^{13} + 6\,q^{12} +
5\,q^{11} + 4\,q^{10} + 5\,q^{9} + 3\,q^{8} + 3\,q^{7} + 3\,q^{6} +
2\,q^{5} + q^{4} + q^{3} ) $\\\\
eigenvalue: $ -q^{5}-q^{3}-q^{2}-1 $\\
multiplicity: $ {1 \over 2} ( q^{21} + q^{20} + q^{18} + 3\,q^{17} +
q^{16} + q^{15} + 4\,q^{14} + 3\,q^{13} + 3\,q^{11} + 4\,q^{10} +
q^{9} + q^{8} + 3\,q^{7} + q^{6} + q^{4}+ q^{3} ) $\\\\

}

\section{$E_{7,1}$ and $E_{7,2}$ and $E_{7,7}$}
The graph $E_{7,1}$ is the Gosset graph, distance-regular with intersection
array $\{27,10,1;\,1,10,27\}$. It has spectrum
$27^1 \z 9^7 \z (-1)^{27} \z (-3)^{21}$.

The graph $E_{7,2}$ has spectrum
$$35^1 \z 25^7 \z 15^{27} \z 7^{56} \z 5^{21} \z 1^{120} \z
(-1)^{35} \z (-3)^{189} \z (-5)^{105} \z (-7)^{15}.$$

The graph $E_{7,7}$ is the root system graph of type $E_7$ on 126 vertices. It has spectrum
$32^1 \z 16^7 \z 4^{27} \z (-2)^{56} \z (-4)^{35}$ (cf. \cite{BCN}, p. 313).\\\\

{\scriptsize

In the thick case $E_{7,1}(q)$ we find a distance-regular graph with spectrum \\

\noindent
eigenvalue: $ q^{17}+q^{16}+q^{15}+q^{14}+2\,q^{13}+2\,q^{12}+2\,q^{11}+2\,q^{10}+3\,q^{9}+2\,q^{8}+2\,q^{7}+2\,q^{6}+2\,q^{5}+q^{4}+q^{3}+q^{2}+q $\\
multiplicity: $ 1 $\\\\
eigenvalue: $ q^{13}+q^{12}+q^{11}+q^{10}+2\,q^{9}+q^{8}+q^{7}+q^{6}+q^{5}-1 $\\
multiplicity: $ q^{17}+q^{13}+q^{11}+q^{9}+q^{7}+q^{5}+q $\\\\
eigenvalue: $ q^{9}-q^{4}-1 $\\
multiplicity: $ q^{26}+q^{24}+2\,q^{22}+2\,q^{20}+3\,q^{18}+3\,q^{16}+3\,q^{14}+3\,q^{12}+3\,q^{10}+2\,q^{8}+2\,q^{6}+q^{4}+q^{2}$\\\\
eigenvalue: $ -q^{8}-q^{4}-1 $\\
multiplicity: $ q^{27}+q^{25}+q^{23}+2\,q^{21}+2\,q^{19}+2\,q^{17}+3\,q^{15}+2\,q^{13}+2\,q^{11}+2\,q^{9}+q^{7}+q^{5}+q^{3} $\\\\

In the thick case $E_{7,2}(q)$ we find \\

\noindent
eigenvalue: $ q^{13}+q^{12}+2\,q^{11}+3\,q^{10}+4\,q^{9}+4\,q^{8}+5\,q^{7}+4\,q^{6}+4\,q^{5}+3\,q^{4}+2\,q^{3}+q^{2}+q $\\
multiplicity: $ 1 $\\\\
eigenvalue: $ q^{11}+2\,q^{10}+3\,q^{9}+4\,q^{8}+5\,q^{7}+4\,q^{6}+4\,q^{5}+2\,q^{4} +q^{3}-1 $\\
multiplicity: $ q^{17}+q^{13}+q^{11}+q^{9}+q^{7}+q^{5}+q $\\\\
eigenvalue: $ q^{10}+2\,q^{9}+3\,q^{8}+4\,q^{7}+3\,q^{6}+3\,q^{5}+q^{4}-q^{2}-1 $\\
multiplicity: $ q^{26}+q^{24}+2\,q^{22}+2\,q^{20}+3\,q^{18}+3\,q^{16}+3\,q^{14}+3\,q^{12}+3\,q^{10}+2\,q^{8}+2\,q^{6}+q^{4}+q^{2} $\\\\
eigenvalue: $ q^{10}+q^{9}+q^{8}+2\,q^{7}+q^{6}+q^{5}-q^{2}-1 $\\
multiplicity: $ q^{27}+q^{25}+q^{23}+2\,q^{21}+2\,q^{19}+2\,q^{17}+3\,q^{15}+2\,q^{13}+2\,q^{11}+2\,q^{9}+q^{7}+q^{5}+q^{3} $\\\\
eigenvalue: $ q^{9}+2\,q^{8}+3\,q^{7}+2\,q^{6}+2\,q^{5}-q^{3}-q^{2}-1 $\\
multiplicity: ${1 \over 2} ( q^{33}+q^{32}+q^{31}+2q^{30}+2q^{29}+3\,q^{28}+3\,q^{27}+3\,q^{26}+4\,q^{25}+5\,q^{24}+5\,q^{23}+5\,q^{22}+6\,q^{21}+6\,q^{20}+6\,q^{19}+6\,q^{18}+6\,q^{17}+6\,q^{16}+6\,q^{15}+5\,q^{14}+5\,q^{13}+5\,q^{12}+4\,q^{11}+3\,q^{10}+3\,q^{9}+3\,q^{8}+2q^{7}+2q^{6}+q^{5}+q^{4}+q^{3} )$\\\\
eigenvalue: $ q^{9}+q^{7}-q^{3}-q^{2}-1 $\\
multiplicity: ${1 \over 2} ( q^{33}+q^{32}+q^{31}+2\,q^{29}+q^{28}+
3\,q^{27}+q^{26}+4\,q^{25}+q^{24}+5\,q^{23}+q^{22}+6\,q^{21}+2\,q^{20}+6\,q^{19}+
6\,q^{17}+2\,q^{16}+6\,q^{15}+q^{14}+5\,q^{13}+q^{12}+4\,q^{11}+q^{10}+3\,q^{9}+
q^{8}+2q^{7}+q^{5}+q^{4}+q^{3} ) $\\\\
eigenvalue: $ q^{8}+2\,q^{7}+q^{6}+q^{5}-q^{4}-q^{3}-q^{2}-1 $\\
multiplicity: ${1 \over 2} ( q^{38}+q^{37}+q^{36}+3\,q^{35}+3\,q^{34}+
3\,q^{33}+5\,q^{32}+6\,q^{31}+6\,q^{30}+8\,q^{29}+9\,q^{28}+9\,q^{27}+
11\,q^{26}+11\,q^{25}+11\,q^{24}+13\,q^{23}+13\,q^{22}+12\,q^{21}+13\,q^{20}+
13\,q^{19}+11\,q^{18}+11\,q^{17}+11\,q^{16}+9\,q^{15}+9\,q^{14}+8\,q^{13}+
6\,q^{12}+6\,q^{11}+5\,q^{10}+3\,q^{9}+3\,q^{8}+3\,q^{7}+q^{6}+q^{5}+q^{4} ) $\\\\
eigenvalue: $ q^{7}-q^{4}-q^{3}-q^{2}-1 $\\
multiplicity: $ q^{41}+2\,q^{39}+4\,q^{37}+6\,q^{35}+9\,q^{33}+12\,q^{31}+15\,q^{29}+17\,q^{27}+19\,q^{25}+19\,q^{23}+19\,q^{21}+17\,q^{19}+15\,q^{17}+12\,q^{15}+9\,q^{13}+6\,q^{11}+4\,q^{9}+2\,q^{7}+q^{5} $\\\\
eigenvalue: $ -q^{6}-q^{4}-q^{3}-q^{2}-1 $\\
multiplicity: $q^{42}+q^{40}+2\,q^{38}+4\,q^{36}+5\,q^{34}+6\,q^{32}+9\,q^{30}+9\,q^{28}+10\,q^{26}+11\,q^{24}+10\,q^{22}+9\,q^{20}+9\,q^{18}+6\,q^{16}+5\,q^{14}+4\,q^{12}+2\,q^{10}+q^{8}+q^{6} $\\\\
eigenvalue: $ -q^{8}-q^{6}-q^{5}-q^{4}-q^{3}-q^{2}-1 $\\
multiplicity: $ {1 \over 2} ( q^{38}-q^{37}+q^{36}-q^{35}+3\,q^{34}-3\,q^{33}+
5\,q^{32}-4\,q^{31}+6\,q^{30}-6\,q^{29}+9\,q^{28}-7\,q^{27}+11\,q^{26}-9\,q^{25}+
11\,q^{24}-9\,q^{23}+13\,q^{22}-10\,q^{21}+13\,q^{20}-9\,q^{19}+11\,q^{18}-9\,q^{17}+
11\,q^{16}-7\,q^{15}+9\,q^{14}-6\,q^{13}+6\,q^{12}-4\,q^{11}+5\,q^{10}-3\,q^{9}+
3\,q^{8}-q^{7}+q^{6}-q^{5}+q^{4} ) $\\\\

In the thick case $E_{7,7}(q)$ we find\\

\noindent
eigenvalue: $ q^{16}+q^{15}+q^{14}+2\,q^{13}+2\,q^{12}+3\,q^{11}+3\,q^{10}+3\,q^{9}+3\,q^{8}+3 \,q^{7}+3\,q^{6}+2\,q^{5}+2\,q^{4}+q^{3}+q^{2}+q $\\
multiplicity: $ 1 $\\\\
eigenvalue: $ q^{13}+q^{12}+2\,q^{11}+2\,q^{10}+2\,q^{9}+3\,q^{8}+2\,q^{7}+2\,q^{6}+q^{5}+q^{4}-1 $\\
multiplicity: $ q^{17}+q^{13}+q^{11}+q^{9}+q^{7}+q^{5}+q $\\\\
eigenvalue: $ q^{11}+q^{10}+q^{9}+q^{8}+q^{7}+q^{6}-q^{3}-1 $\\
multiplicity: $q^{26}+q^{24}+2\,q^{22}+2\,q^{20}+3\,q^{18}+3\,q^{16}+3\,q^{14}+3\,q^{12}+3\,q^{10}+2\,q^{8}+2\,q^{6}+q^{4}+q^{2} $\\\\
eigenvalue: $ q^{8}-q^{5}-q^{3}-1 $\\
multiplicity: $ {1 \over 2} ( q^{33}+q^{32}+q^{31}+2\,q^{30}+2\,q^{29}+
3\,q^{28}+3\,q^{27}+3\,q^{26}+4\,q^{25}+5\,q^{24}+5\,q^{23}+5\,q^{22}+6\,q^{21}+
6\,q^{20}+6\,q^{19}+6\,q^{18}+6\,q^{17}+6\,q^{16}+6\,q^{15}+5\,q^{14}+5\,q^{13}+
5\,q^{12}+4\,q^{11}+3\,q^{10}+3\,q^{9}+3\,q^{8}+2\,q^{7}+2\,q^{6}+q^{5}+q^{4}+q^{3})$\\\\
eigenvalue: $ -q^{8}-q^{5}-q^{3}-1 $\\
multiplicity: $ {1 \over 2} ( q^{33}+q^{32}+q^{31}+2\,q^{29}+q^{28}+3\,q^{27}+
q^{26}+4\,q^{25}+q^{24}+5\,q^{23}+q^{22}+6\,q^{21}+2\,q^{20}+6\,q^{19}+6\,q^{17}+
2\,q^{16}+6\,q^{15}+q^{14}+5\,q^{13}+q^{12}+4\,q^{11}+q^{10}+3\,q^{9}+
q^{8}+2\,q^{7}+q^{5}+q^{4}+q^{3} ) $\\\\

}

\section{$E_{8,1}$ and $E_{8,2}$ and $E_{8,8}$}
The graph $E_{8,1}$ has spectrum
$56^1 \z 28^8 \z 8^{35} \z (-2)^{112} \z (-4)^{84}$.

The graph $E_{8,2}$ has spectrum
$$56^1 \z 49^8 \z 41^{35} \z {1 \over 2}(45 + {\sqrt {409}})^{112} \z
25^{210} \z {1 \over 2}(25 + 3{\sqrt {41}})^{84} \z 17^{560} \z
{1 \over 2}(45 - {\sqrt {409}})^{112}$$
$${1 \over 2}(9 + {\sqrt {145}})^{700} \z
9^{567} \z 5^{1400} \z 4^{400} \z {1 \over 2}(25 - 3{\sqrt {41}})^{84} \z
1^{1960} \z (-1)^{1344}$$
$${1 \over 2}(9 - {\sqrt {145}})^{700} \z
(-3)^{3240} \z (-4)^{448} \z (-5)^{2240} \z (-7)^{2900} \z (-8)^{175}.$$
(Here the nonintegral eigenvalues are approximately
32.611874, 22.104686, 12.388126, 10.520797, 2.895314 and $-1.520797$.
In this case the association scheme has $d+1 = 35$ relations, while
$L_1$ has only 21 distinct eigenvalues. Indeed, the nonintegral
eigenvalues and 17 and 4 all have multiplicity 2 in $L_1$, and
$1$ and $-7$ have multiplicities 3 and 5, respectively.)

The graph $E_{8,8}$ is the root system graph of type $E_8$ on 2160 vertices. It has spectrum
$$64^1 \z 48^8 \z 32^{35} \z 18^{112} \z 8^{210} \z 4^{84} \z
0^{560} \z (-4)^{700} \z (-6)^{400} \z (-8)^{50}$$ (cf. \cite{BCN}, pp. 313-314).\\\\

{\scriptsize

In the thick case $E_{8,1}(q)$ we find\\

\noindent
eigenvalue: $ q^{28}+q^{27}+q^{26}+q^{25}+q^{24}+2\,q^{23}+2\,q^{22}+2\,q^{21}+2\,q^{20}+3\,q^{19}+3\,q^{18}+3\,q^{17}+3\,q^{16}+3\,q^{15}+3\,q^{14}+3\,q^{13}+3\,q^{12}+3\,q^{11}+3\,q^{10}+2\,q^{9}+2\,q^{8}+2\,q^{7}+2\,q^{6}+q^{5}+q^{4}+q^{3}+q^{2}+q $\\
multiplicity: $ 1 $\\\\
eigenvalue: $ q^{23}+q^{22}+q^{21}+q^{20}+2\,q^{19}+2\,q^{18}+2\,q^{17}+2\,q^{16}+2 \,q^{15}+3\,q^{14}+2\,q^{13}+2\,q^{12}+2\,q^{11}+2\,q^{10}+q^{9}+q^{8} +q^{7}+q^{6}-1 $\\
multiplicity: $ q^{29}+q^{23}+q^{19}+q^{17}+q^{13}+q^{11}+q^{7}+q $\\\\
eigenvalue: $ q^{19}+q^{18}+q^{17}+q^{16}+q^{15}+q^{14}+q^{13}+q^{12}+q^{11}+q^{10}-q^{5}-1 $\\
multiplicity: $q^{46}+q^{42}+q^{40}+q^{38}+2\,q^{36}+2\,q^{34}+q^{32}+3\,q^{30}+2\,q^{28}+2\,q^{26}+3\,q^{24}+2\,q^{22}+2\,q^{20}+3\,q^{18}+q^{16}+2\,q^{14}+2\,q^{12}+q^{10}+q^{8}+q^{6}+q^{2} $\\\\
eigenvalue: $ q^{14}-q^{9}-q^{5}-1 $\\
multiplicity: ${1 \over 2} ( q^{57}+q^{56}+q^{55}+q^{54}+q^{53}+2\,q^{52}+
2\,q^{51}+2\,q^{50}+2\,q^{49}+3\,q^{48}+3\,q^{47}+3\,q^{46}+4\,q^{45}+4\,q^{44}+
4\,q^{43}+5\,q^{42}+5\,q^{41}+6\,q^{40}+6\,q^{39}+5\,q^{38}+6\,q^{37}+
7\,q^{36}+7\,q^{35}+6\,q^{34}+7\,q^{33}+7\,q^{32}+7\,q^{31}+8\,q^{30}+
7\,q^{29}+7\,q^{28}+7\,q^{27}+6\,q^{26}+7\,q^{25}+7\,q^{24}+6\,q^{23}+
5\,q^{22}+6\,q^{21}+6\,q^{20}+5\,q^{19}+5\,q^{18}+4\,q^{17}+4\,q^{16}+
4\,q^{15}+3\,q^{14}+3\,q^{13}+3\,q^{12}+2\,q^{11}+2\,q^{10}+2\,q^{9}+
2\,q^{8}+q^{7}+q^{6}+q^{5}+q^{4}+q^{3} )$\\\\
eigenvalue: $ -q^{14}-q^{9}-q^{5}-1 $\\
multiplicity: ${1 \over 2} ( q^{57}+q^{56}+q^{55}+q^{54}+q^{53}+2\,q^{51}+
2\,q^{50}+2\,q^{49}+q^{48}+3\,q^{47}+q^{46}+4\,q^{45}+4\,q^{44}+4\,q^{43}+
q^{42}+5\,q^{41}+2\,q^{40}+6\,q^{39}+5\,q^{38}+6\,q^{37}+q^{36}+7\,q^{35}+
4\,q^{34}+7\,q^{33}+5\,q^{32}+7\,q^{31}+7\,q^{29}+5\,q^{28}+7\,q^{27}+
4\,q^{26}+7\,q^{25}+q^{24}+6\,q^{23}+5\,q^{22}+6\,q^{21}+2\,q^{20}+5\,q^{19}+
q^{18}+4\,q^{17}+4\,q^{16}+4\,q^{15}+q^{14}+3\,q^{13}+q^{12}+2\,q^{11}+
2\,q^{10}+2\,q^{9}+q^{7}+q^{6}+q^{5}+q^{4}+q^{3} ) $\\\\


In the thick case $E_{8,8}(q)$ we find \\

\noindent
eigenvalue: $ q^{22}+q^{21}+q^{20}+2\,q^{19}+2\,q^{18}+3\,q^{17}+4\,q^{16}+4\,q^{15}+4\,q^{14}+5\,q^{13}+5\,q^{12}+5\,q^{11}+5\,q^{10}+4\,q^{9}+4\,q^{8}+4\,q^{7}+3\,q^{6}+2\,q^{5}+2\,q^{4}+q^{3}+q^{2}+q $\\
multiplicity: $ 1 $\\\\
eigenvalue: $ q^{19}+q^{18}+2\,q^{17}+3\,q^{16}+3\,q^{15}+4\,q^{14}+5\,q^{13}+5\,q^{12}+5\,q^{11}+5\,q^{10}+4\,q^{9}+4\,q^{8}+3\,q^{7}+2\,q^{6}+q^{5}+q^{4}-1 $\\
multiplicity: $q^{29}+q^{23}+q^{19}+q^{17}+q^{13}+q^{11}+q^{7}+q $\\\\
eigenvalue: $ q^{17}+2\,q^{16}+2\,q^{15}+3\,q^{14}+4\,q^{13}+4\,q^{12}+5\,q^{11}+4\,q^{10}+3\,q^{9}+3\,q^{8}+2\,q^{7}+q^{6}-q^{3}-1 $\\
multiplicity: $ q^{46}+q^{42}+q^{40}+q^{38}+2\,q^{36}+2\,q^{34}+q^{32}+3\,q^{30}+2\,q^{28}+2\,q^{26}+3\,q^{24}+2\,q^{22}+2\,q^{20}+3\,q^{18}+q^{16}+2\,q^{14}+2\,q^{12}+q^{10}+q^{8}+q^{6}+q^{2} $\\\\
eigenvalue: $ q^{16}+q^{15}+2\,q^{14}+3\,q^{13}+3\,q^{12}+3\,q^{11}+3\,q^{10}+2\,q^{9}+2\,q^{8}+q^{7}-q^{5}-q^{3}-1 $\\
multiplicity: ${1 \over 2} ( q^{57}+q^{56}+q^{55}+q^{54}+q^{53}+2\,q^{52}+
2\,q^{51}+2\,q^{50}+2\,q^{49}+3\,q^{48}+3\,q^{47}+3\,q^{46}+4\,q^{45}+
4\,q^{44}+4\,q^{43}+5\,q^{42}+5\,q^{41}+6\,q^{40}+6\,q^{39}+5\,q^{38}+
6\,q^{37}+7\,q^{36}+7\,q^{35}+6\,q^{34}+7\,q^{33}+7\,q^{32}+7\,q^{31}+
8\,q^{30}+7\,q^{29}+7\,q^{28}+7\,q^{27}+6\,q^{26}+7\,q^{25}+7\,q^{24}+
6\,q^{23}+5\,q^{22}+6\,q^{21}+6\,q^{20}+5\,q^{19}+5\,q^{18}+4\,q^{17}+
4\,q^{16}+4\,q^{15}+3\,q^{14}+3\,q^{13}+3\,q^{12}+2\,q^{11}+2\,q^{10}+
2\,q^{9}+2\,q^{8}+q^{7}+q^{6}+q^{5}+q^{4}+q^{3} ) $\\\\
eigenvalue: $ q^{16}+q^{15}+q^{13}+q^{12}+q^{11}+q^{10}+q^{7}-q^{5}-q^{3}-1 $\\
multiplicity: ${1 \over 2} ( q^{57}+q^{56}+q^{55}+q^{54}+q^{53}+2\,q^{51}+
2\,q^{50}+2\,q^{49}+q^{48}+3\,q^{47}+q^{46}+4\,q^{45}+4\,q^{44}+4\,q^{43}+q^{42}+
5\,q^{41}+2\,q^{40}+6\,q^{39}+5\,q^{38}+6\,q^{37}+q^{36}+7\,q^{35}+4\,q^{34}+
7\,q^{33}+5\,q^{32}+7\,q^{31}+7\,q^{29}+5\,q^{28}+7\,q^{27}+4\,q^{26}+7\,q^{25}+
q^{24}+6\,q^{23}+5\,q^{22}+6\,q^{21}+2\,q^{20}+5\,q^{19}+q^{18}+4\,q^{17}+
4\,q^{16}+4\,q^{15}+q^{14}+3\,q^{13}+q^{12}+2\,q^{11}+2\,q^{10}+2\,q^{9}+
q^{7}+q^{6}+q^{5}+q^{4}+q^{3} ) $\\\\
eigenvalue: $ q^{14}+2\,q^{13}+2\,q^{12}+3\,q^{11}+2\,q^{10}+q^{9}+q^{8}-q^{6}-q^{5}-q^{3}-1 $\\
multiplicity: ${1 \over 2} ( q^{68}+q^{65}+2\,q^{64}+q^{63}+2\,q^{62}+
q^{61}+3\,q^{60}+3\,q^{59}+4\,q^{58}+2\,q^{57}+6\,q^{56}+3\,q^{55}+5\,q^{54}+
6\,q^{53}+9\,q^{52}+4\,q^{51}+9\,q^{50}+6\,q^{49}+11\,q^{48}+9\,q^{47}+
12\,q^{46}+6\,q^{45}+14\,q^{44}+9\,q^{43}+13\,q^{42}+11\,q^{41}+16\,q^{40}+7\,q^{39}+
15\,q^{38}+11\,q^{37}+16\,q^{36}+11\,q^{35}+15\,q^{34}+7\,q^{33}+16\,q^{32}+
11\,q^{31}+13\,q^{30}+9\,q^{29}+14\,q^{28}+6\,q^{27}+12\,q^{26}+9\,q^{25}+
11\,q^{24}+6\,q^{23}+9\,q^{22}+4\,q^{21}+9\,q^{20}+6\,q^{19}+5\,q^{18}+
3\,q^{17}+6\,q^{16}+2\,q^{15}+4\,q^{14}+3\,q^{13}+3\,q^{12}+q^{11}+
2\,q^{10}+q^{9}+2\,q^{8}+q^{7}+q^{4} ) $\\\\
eigenvalue: $ q^{13}+q^{12}+q^{11}+q^{10}-q^{6}-q^{5}-q^{3}-1 $\\
multiplicity: $q^{73}+q^{71}+2\,q^{69}+4\,q^{67}+4\,q^{65}+7\,q^{63}+10\,q^{61}+10\,q^{59}+15\,q^{57}+18\,q^{55}+18\,q^{53}+25\,q^{51}+26\,q^{49}+26\,q^{47}+33\,q^{45}+31\,q^{43}+31\,q^{41}+36\,q^{39}+31\,q^{37}+31\,q^{35}+33\,q^{33}+26\,q^{31}+26\,q^{29}+25\,q^{27}+18\,q^{25}+18\,q^{23}+15\,q^{21}+10\,q^{19}+10\,q^{17}+7\,q^{15}+4\,q^{13}+4\,q^{11}+2\,q^{9}+q^{7}+q^{5} $\\\\
eigenvalue: $ q^{11}-q^{9}-q^{6}-q^{5}-q^{3}-1 $\\
multiplicity: $ {1 \over 2} ( q^{78}+q^{77}+2\,q^{76}+q^{75}+2\,q^{74}+
q^{73}+5\,q^{72}+3\,q^{71}+7\,q^{70}+3\,q^{69}+8\,q^{68}+4\,q^{67}+14\,q^{66}+
7\,q^{65}+17\,q^{64}+7\,q^{63}+19\,q^{62}+9\,q^{61}+28\,q^{60}+13\,q^{59}+
31\,q^{58}+12\,q^{57}+33\,q^{56}+15\,q^{55}+44\,q^{54}+19\,q^{53}+45\,q^{52}+
17\,q^{51}+47\,q^{50}+21\,q^{49}+57\,q^{48}+23\,q^{47}+54\,q^{46}+20\,q^{45}+
55\,q^{44}+24\,q^{43}+62\,q^{42}+24\,q^{41}+55\,q^{40}+20\,q^{39}+54\,q^{38}+
23\,q^{37}+57\,q^{36}+21\,q^{35}+47\,q^{34}+17\,q^{33}+45\,q^{32}+
19\,q^{31}+44\,q^{30}+15\,q^{29}+33\,q^{28}+12\,q^{27}+31\,q^{26}+
13\,q^{25}+28\,q^{24}+9\,q^{23}+19\,q^{22}+7\,q^{21}+17\,q^{20}+7\,q^{19}+
14\,q^{18}+4\,q^{17}+8\,q^{16}+3\,q^{15}+7\,q^{14}+3\,q^{13}+5\,q^{12}+
q^{11}+2\,q^{10}+q^{9}+2\,q^{8}+q^{7}+q^{6} ) $\\\\
eigenvalue: $ -q^{11}-q^{9}-q^{6}-q^{5}-q^{3}-1 $\\
multiplicity: ${1 \over 2} ( q^{78}+q^{77}+q^{75}+2\,q^{74}+q^{73}+
q^{72}+3\,q^{71}+3\,q^{70}+3\,q^{69}+4\,q^{68}+4\,q^{67}+6\,q^{66}+
7\,q^{65}+5\,q^{64}+7\,q^{63}+11\,q^{62}+9\,q^{61}+8\,q^{60}+13\,q^{59}+
13\,q^{58}+12\,q^{57}+15\,q^{56}+15\,q^{55}+16\,q^{54}+19\,q^{53}+17\,q^{52}+
17\,q^{51}+23\,q^{50}+21\,q^{49}+17\,q^{48}+23\,q^{47}+24\,q^{46}+20\,q^{45}+
23\,q^{44}+24\,q^{43}+22\,q^{42}+24\,q^{41}+23\,q^{40}+20\,q^{39}+24\,q^{38}+
23\,q^{37}+17\,q^{36}+21\,q^{35}+23\,q^{34}+17\,q^{33}+17\,q^{32}+19\,q^{31}+
16\,q^{30}+15\,q^{29}+15\,q^{28}+12\,q^{27}+13\,q^{26}+13\,q^{25}+8\,q^{24}+
9\,q^{23}+11\,q^{22}+7\,q^{21}+5\,q^{20}+7\,q^{19}+6\,q^{18}+4\,q^{17}+4\,q^{16}+
3\,q^{15}+3\,q^{14}+3\,q^{13}+q^{12}+q^{11}+2\,q^{10}+q^{9}+q^{7}+q^{6} ) $\\\\
eigenvalue: $ -q^{14}-q^{11}-q^{9}-q^{8}-q^{6}-q^{5}-q^{3}-1 $\\
multiplicity: $ {1 \over 2} (q^{68}-q^{65}+2\,q^{64}-q^{63}+2\,q^{62}-q^{61}+
3\,q^{60}-3\,q^{59}+4\,q^{58}-2\,q^{57}+6\,q^{56}-3\,q^{55}+5\,q^{54}-6\,q^{53}+
9\,q^{52}-4\,q^{51}+9\,q^{50}-6\,q^{49}+11\,q^{48}-9\,q^{47}+12\,q^{46}-6\,q^{45}+
14\,q^{44}-9\,q^{43}+13\,q^{42}-11\,q^{41}+16\,q^{40}-7\,q^{39}+15\,q^{38}-11\,q^{37}+
16\,q^{36}-11\,q^{35}+15\,q^{34}-7\,q^{33}+16\,q^{32}-11\,q^{31}+13\,q^{30}-9\,q^{29}+
14\,q^{28}-6\,q^{27}+12\,q^{26}-9\,q^{25}+11\,q^{24}-6\,q^{23}+9\,q^{22}-4\,q^{21}+
9\,q^{20}-6\,q^{19}+5\,q^{18}-3\,q^{17}+6\,q^{16}-2\,q^{15}+4\,q^{14}-3\,q^{13}+
3\,q^{12}-q^{11}+2\,q^{10}-q^{9}+2\,q^{8}-q^{7}+q^{4} ) $\\\\

In the thick case $E_{8,2}(q)$ we find\\

\noindent
eigenvalue: $q^{16}+q^{15}+2\,q^{14}+3\,q^{13}+4\,q^{12}+5\,q^{11}+6\,q^{10}+6\,q^{9}+6\,q^{8}+6\,q^{7}+5\,q^{6}+4\,q^{5}+3\,q^{4}+2\,q^{3}+q^{2}+q$\\
multiplicity: $1$\\\\
eigenvalue: $q^{14}+2\,q^{13}+3\,q^{12}+5\,q^{11}+6\,q^{10}+7\,q^{9}+7\,q^{8}+7\,q^{7}+5\,q^{6}+4\,q^{5}+2\,q^{4}+q^{3}-1$\\
multiplicity: $q^{29}+q^{23}+q^{19}+q^{17}+q^{13}+q^{11}+q^{7}+q$\\\\
eigenvalue: $q^{13}+2\,q^{12}+4\,q^{11}+6\,q^{10}+7\,q^{9}+7\,q^{8}+7\,q^{7}+5\,q^{6}+3\,q^{5}+q^{4}-q^{2}-1$\\
multiplicity: $q^{46}+q^{42}+q^{40}+q^{38}+2\,q^{36}+2\,q^{34}+q^{32}+3\,q^{30}+2\,q^{28}+2\,q^{26}+3\,q^{24}+2\,q^{22}+2\,q^{20}+3\,q^{18}+q^{16}+2\,q^{14}+2\,q^{12}+q^{10}+q^{8}+q^{6}+q^{2}$\\\\
eigenvalue: $2\,q^{11}+4\,q^{10}+6\,q^{9}+7\,q^{8}+6\,q^{7}+3\,q^{6}+q^{5}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: ${ 1 \over 2} ( q^{68}+q^{65}+2\,q^{64}+q^{63}+2\,q^{62}+q^{61}+3\,q^{60}+3\,q^{59}+4
\,q^{58}+2\,q^{57}+6\,q^{56}+3\,q^{55}+5\,q^{54}+6\,q^{53}+9\,q^{52}+4
\,q^{51}+9\,q^{50}+6\,q^{49}+11\,q^{48}+9\,q^{47}+12\,q^{46}+6\,q^{45}
+14\,q^{44}+9\,q^{43}+13\,q^{42}+11\,q^{41}+16\,q^{40}+7\,q^{39}+15\,q
^{38}+11\,q^{37}+16\,q^{36}+11\,q^{35}+15\,q^{34}+7\,q^{33}+16\,q^{32}
+11\,q^{31}+13\,q^{30}+9\,q^{29}+14\,q^{28}+6\,q^{27}+12\,q^{26}+9\,q^
{25}+11\,q^{24}+6\,q^{23}+9\,q^{22}+4\,q^{21}+9\,q^{20}+6\,q^{19}+5\,q
^{18}+3\,q^{17}+6\,q^{16}+2\,q^{15}+4\,q^{14}+3\,q^{13}+3\,q^{12}+q^{
11}+2\,q^{10}+q^{9}+2\,q^{8}+q^{7}+q^{4})$\\\\
eigenvalue: $q^{11}+3\,q^{10}+5\,q^{9}+5\,q^{8}+5\,q^{7}+2\,q^{6}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: $q^{73}+q^{71}+2\,q^{69}+4\,q^{67}+4\,q^{65}+7\,q^{63}+10\,q^{61}+10\,q^{59}+15\,q^{57}+18\,q^{55}+18\,q^{53}+25\,q^{51}+26\,q^{49}+26\,q^{47}+33\,q^{45}+31\,q^{43}+31\,q^{41}+36\,q^{39}+31\,q^{37}+31\,q^{35}+33\,q^{33}+26\,q^{31}+26\,q^{29}+25\,q^{27}+18\,q^{25}+18\,q^{23}+15\,q^{21}+10\,q^{19}+10\,q^{17}+7\,q^{15}+4\,q^{13}+4\,q^{11}+2\,q^{9}+q^{7}+q^{5}$\\\\
eigenvalue: $q^{11}+2\,q^{10}+3\,q^{9}+3\,q^{8}+3\,q^{7}+q^{6}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: $q^{74}+q^{72}+2\,q^{70}+3\,q^{68}+5\,q^{66}+7\,q^{64}+9\,q^{62}+11\,q^{60}+15\,q^{58}+17\,q^{56}+21\,q^{54}+23\,q^{52}+26\,q^{50}+29\,q^{48}+31\,q^{46}+32\,q^{44}+34\,q^{42}+33\,q^{40}+34\,q^{38}+32\,q^{36}+31\,q^{34}+29\,q^{32}+26\,q^{30}+23\,q^{28}+21\,q^{26}+17\,q^{24}+15\,q^{22}+11\,q^{20}+9\,q^{18}+7\,q^{16}+5\,q^{14}+3\,q^{12}+2\,q^{10}+q^{8}+q^{6}$\\\\
eigenvalue: $q^{11}+q^{10}+2\,q^{9}+2\,q^{8}+2\,q^{7}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: ${ 1 \over 2} ( q^{78}+q^{77}+q^{75}+2\,q^{74}+q^{73}+q^{72}+3\,q^{71}+3\,q^{70}+3\,q^
{69}+4\,q^{68}+4\,q^{67}+6\,q^{66}+7\,q^{65}+5\,q^{64}+7\,q^{63}+11\,q
^{62}+9\,q^{61}+8\,q^{60}+13\,q^{59}+13\,q^{58}+12\,q^{57}+15\,q^{56}+
15\,q^{55}+16\,q^{54}+19\,q^{53}+17\,q^{52}+17\,q^{51}+23\,q^{50}+21\,
q^{49}+17\,q^{48}+23\,q^{47}+24\,q^{46}+20\,q^{45}+23\,q^{44}+24\,q^{
43}+22\,q^{42}+24\,q^{41}+23\,q^{40}+20\,q^{39}+24\,q^{38}+23\,q^{37}+
17\,q^{36}+21\,q^{35}+23\,q^{34}+17\,q^{33}+17\,q^{32}+19\,q^{31}+16\,
q^{30}+15\,q^{29}+15\,q^{28}+12\,q^{27}+13\,q^{26}+13\,q^{25}+8\,q^{24
}+9\,q^{23}+11\,q^{22}+7\,q^{21}+5\,q^{20}+7\,q^{19}+6\,q^{18}+4\,q^{
17}+4\,q^{16}+3\,q^{15}+3\,q^{14}+3\,q^{13}+q^{12}+q^{11}+2\,q^{10}+q^
{9}+q^{7}+q^{6} )$\\\\
eigenvalue: $q^{11}+q^{10}+q^{9}+q^{8}+q^{7}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: $q^{73}+q^{71}+2\,q^{69}+4\,q^{67}+4\,q^{65}+7\,q^{63}+10\,q^{61}+10\,q^{59}+15\,q^{57}+18\,q^{55}+18\,q^{53}+25\,q^{51}+26\,q^{49}+26\,q^{47}+33\,q^{45}+31\,q^{43}+31\,q^{41}+36\,q^{39}+31\,q^{37}+31\,q^{35}+33\,q^{33}+26\,q^{31}+26\,q^{29}+25\,q^{27}+18\,q^{25}+18\,q^{23}+15\,q^{21}+10\,q^{19}+10\,q^{17}+7\,q^{15}+4\,q^{13}+4\,q^{11}+2\,q^{9}+q^{7}+q^{5}$\\\\
eigenvalue: $q^{10}+3\,q^{9}+3\,q^{8}+3\,q^{7}-q^{5}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: ${ 1 \over 6} ( q^{83}+3\,q^{82}+5\,q^{81}+6\,q^{80}+8\,q^{79}+9\,q^{78}+11\,q^{77}+15 \,q^{76}+22\,q^{75}+24\,q^{74}+29\,q^{73}+33\,q^{72}+40\,q^{71}+45\,q^ {70}+60\,q^{69}+63\,q^{68}+71\,q^{67}+75\,q^{66}+90\,q^{65}+93\,q^{64} +117\,q^{63}+120\,q^{62}+132\,q^{61}+132\,q^{60}+157\,q^{59}+153\,q^{
58}+181\,q^{57}+180\,q^{56}+194\,q^{55}+183\,q^{54}+217\,q^{53}+207\,q
^{52}+233\,q^{51}+225\,q^{50}+241\,q^{49}+216\,q^{48}+249\,q^{47}+234
\,q^{46}+252\,q^{45}+234\,q^{44}+249\,q^{43}+216\,q^{42}+241\,q^{41}+
225\,q^{40}+233\,q^{39}+207\,q^{38}+217\,q^{37}+183\,q^{36}+194\,q^{35
}+180\,q^{34}+181\,q^{33}+153\,q^{32}+157\,q^{31}+132\,q^{30}+132\,q^{
29}+120\,q^{28}+117\,q^{27}+93\,q^{26}+90\,q^{25}+75\,q^{24}+71\,q^{23
}+63\,q^{22}+60\,q^{21}+45\,q^{20}+40\,q^{19}+33\,q^{18}+29\,q^{17}+24
\,q^{16}+22\,q^{15}+15\,q^{14}+11\,q^{13}+9\,q^{12}+8\,q^{11}+6\,q^{10
}+5\,q^{9}+3\,q^{8}+q^{7} )$\\\\
eigenvalue: $q^{10}+q^{9}+q^{8}+q^{7}-q^{5}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: ${ 1 \over 2} (q^{83}+q^{82}+q^{81}+2\,q^{80}+2\,q^{79}+3\,q^{78}+5\,q^{77}+5\,q^{76}
+6\,q^{75}+8\,q^{74}+9\,q^{73}+11\,q^{72}+14\,q^{71}+15\,q^{70}+16\,q^
{69}+21\,q^{68}+23\,q^{67}+25\,q^{66}+30\,q^{65}+31\,q^{64}+33\,q^{63}
+40\,q^{62}+42\,q^{61}+44\,q^{60}+49\,q^{59}+51\,q^{58}+53\,q^{57}+60
\,q^{56}+62\,q^{55}+61\,q^{54}+67\,q^{53}+69\,q^{52}+69\,q^{51}+75\,q^
{50}+75\,q^{49}+72\,q^{48}+77\,q^{47}+78\,q^{46}+76\,q^{45}+78\,q^{44}
+77\,q^{43}+72\,q^{42}+75\,q^{41}+75\,q^{40}+69\,q^{39}+69\,q^{38}+67
\,q^{37}+61\,q^{36}+62\,q^{35}+60\,q^{34}+53\,q^{33}+51\,q^{32}+49\,q^
{31}+44\,q^{30}+42\,q^{29}+40\,q^{28}+33\,q^{27}+31\,q^{26}+30\,q^{25}
+25\,q^{24}+23\,q^{23}+21\,q^{22}+16\,q^{21}+15\,q^{20}+14\,q^{19}+11
\,q^{18}+9\,q^{17}+8\,q^{16}+6\,q^{15}+5\,q^{14}+5\,q^{13}+3\,q^{12}+2
\,q^{11}+2\,q^{10}+q^{9}+q^{8}+q^{7} )$\\\\
eigenvalue: $q^{10}-q^{5}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: ${ 1 \over 3} (q^{83}+2\,q^{81}+2\,q^{79}+5\,q^{77}+7\,q^{75}+8\,q^{73}+16\,q^{71}+18
\,q^{69}+20\,q^{67}+33\,q^{65}+33\,q^{63}+39\,q^{61}+55\,q^{59}+49\,q^
{57}+59\,q^{55}+73\,q^{53}+62\,q^{51}+76\,q^{49}+81\,q^{47}+66\,q^{45}
+81\,q^{43}+76\,q^{41}+62\,q^{39}+73\,q^{37}+59\,q^{35}+49\,q^{33}+55
\,q^{31}+39\,q^{29}+33\,q^{27}+33\,q^{25}+20\,q^{23}+18\,q^{21}+16\,q^
{19}+8\,q^{17}+7\,q^{15}+5\,q^{13}+2\,q^{11}+2\,q^{9}+q^{7} )$\\\\
eigenvalue: $2\,q^{9}+3\,q^{8}+2\,q^{7}-q^{6}-q^{5}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: ${1 \over 6} (q^{88}+5\,q^{86}+11\,q^{84}+17\,q^{82}+27\,q^{80}+43\,q^{78}+61\,q^{76
}+84\,q^{74}+114\,q^{72}+145\,q^{70}+180\,q^{68}+220\,q^{66}+258\,q^{
64}+300\,q^{62}+344\,q^{60}+377\,q^{58}+408\,q^{56}+439\,q^{54}+457\,q
^{52}+470\,q^{50}+478\,q^{48}+470\,q^{46}+457\,q^{44}+439\,q^{42}+408
\,q^{40}+377\,q^{38}+344\,q^{36}+300\,q^{34}+258\,q^{32}+220\,q^{30}+
180\,q^{28}+145\,q^{26}+114\,q^{24}+84\,q^{22}+61\,q^{20}+43\,q^{18}+
27\,q^{16}+17\,q^{14}+11\,q^{12}+5\,q^{10}+q^{8} ) $\\\\
eigenvalue: $q^{9}+q^{8}+q^{7}-q^{6}-q^{5}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: $q^{89}+2\,q^{87}+4\,q^{85}+7\,q^{83}+11\,q^{81}+17\,q^{79}+24\,q^{77}+33\,q^{75}+44\,q^{73}+56\,q^{71}+70\,q^{69}+85\,q^{67}+100\,q^{65}+116\,q^{63}+131\,q^{61}+145\,q^{59}+158\,q^{57}+168\,q^{55}+176\,q^{53}+
181\,q^{51}+182\,q^{49}+181\,q^{47}+176\,q^{45}+168\,q^{43}+158\,q^{41}+145\,q^{39}+131\,q^{37}+116\,q^{35}+100\,q^{33}+85\,q^{31}+70\,q^{29}+56\,q^{27}+44\,q^{25}+33\,q^{23}+24\,q^{21}+17\,q^{19}+11\,q^{17}+7\,q^{15}+4\,q^{13}+2\,q^{11}+q^{9}$\\\\
eigenvalue: $q^{8}-q^{6}-q^{5}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: ${ 1 \over 2} (q^{92}+q^{91}+q^{90}+2\,q^{89}+2\,q^{88}+3\,q^{87}+5\,q^{86}+5\,q^{85}
+6\,q^{84}+9\,q^{83}+10\,q^{82}+12\,q^{81}+16\,q^{80}+17\,q^{79}+19\,q
^{78}+25\,q^{77}+27\,q^{76}+30\,q^{75}+37\,q^{74}+39\,q^{73}+42\,q^{72
}+51\,q^{71}+54\,q^{70}+57\,q^{69}+66\,q^{68}+69\,q^{67}+72\,q^{66}+83
\,q^{65}+86\,q^{64}+87\,q^{63}+97\,q^{62}+100\,q^{61}+101\,q^{60}+111
\,q^{59}+113\,q^{58}+111\,q^{57}+120\,q^{56}+122\,q^{55}+119\,q^{54}+
126\,q^{53}+126\,q^{52}+120\,q^{51}+126\,q^{50}+126\,q^{49}+119\,q^{48
}+122\,q^{47}+120\,q^{46}+111\,q^{45}+113\,q^{44}+111\,q^{43}+101\,q^{
42}+100\,q^{41}+97\,q^{40}+87\,q^{39}+86\,q^{38}+83\,q^{37}+72\,q^{36}
+69\,q^{35}+66\,q^{34}+57\,q^{33}+54\,q^{32}+51\,q^{31}+42\,q^{30}+39
\,q^{29}+37\,q^{28}+30\,q^{27}+27\,q^{26}+25\,q^{25}+19\,q^{24}+17\,q^
{23}+16\,q^{22}+12\,q^{21}+10\,q^{20}+9\,q^{19}+6\,q^{18}+5\,q^{17}+5
\,q^{16}+3\,q^{15}+2\,q^{14}+2\,q^{13}+q^{12}+q^{11}+q^{10})$\\\\
eigenvalue: $-q^{8}-q^{6}-q^{5}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: ${ 1 \over 2} (q^{92}+q^{91}+q^{90}+3\,q^{88}+q^{87}+6\,q^{86}+3\,q^{85}+9\,q^{84}+q^
{83}+15\,q^{82}+4\,q^{81}+23\,q^{80}+7\,q^{79}+31\,q^{78}+4\,q^{77}+44
\,q^{76}+11\,q^{75}+59\,q^{74}+14\,q^{73}+73\,q^{72}+10\,q^{71}+94\,q^
{70}+22\,q^{69}+115\,q^{68}+23\,q^{67}+134\,q^{66}+19\,q^{65}+159\,q^{
64}+35\,q^{63}+182\,q^{62}+32\,q^{61}+200\,q^{60}+29\,q^{59}+223\,q^{
58}+47\,q^{57}+241\,q^{56}+38\,q^{55}+251\,q^{54}+37\,q^{53}+265\,q^{
52}+53\,q^{51}+272\,q^{50}+39\,q^{49}+269\,q^{48}+40\,q^{47}+270\,q^{
46}+51\,q^{45}+263\,q^{44}+34\,q^{43}+247\,q^{42}+37\,q^{41}+236\,q^{
40}+42\,q^{39}+218\,q^{38}+25\,q^{37}+193\,q^{36}+29\,q^{35}+176\,q^{
34}+29\,q^{33}+153\,q^{32}+15\,q^{31}+127\,q^{30}+19\,q^{29}+110\,q^{
28}+16\,q^{27}+89\,q^{26}+7\,q^{25}+68\,q^{24}+10\,q^{23}+56\,q^{22}+7
\,q^{21}+41\,q^{20}+2\,q^{19}+28\,q^{18}+4\,q^{17}+22\,q^{16}+2\,q^{15
}+14\,q^{14}+8\,q^{12}+q^{11}+6\,q^{10}+3\,q^{8}+q^{6}+q^{4})$\\\\
eigenvalue: $-q^{9}-q^{7}-q^{6}-q^{5}-q^{4}-q^{3}-q^{2}-1$\\
multiplicity: ${ 1 \over 3} (q^{88}-q^{86}+2\,q^{84}+2\,q^{82}+4\,q^{78}+7\,q^{76}+12\,q^{72}+10\,q
^{70}+6\,q^{68}+19\,q^{66}+18\,q^{64}+9\,q^{62}+32\,q^{60}+20\,q^{58}+
18\,q^{56}+37\,q^{54}+25\,q^{52}+20\,q^{50}+43\,q^{48}+20\,q^{46}+25\,
q^{44}+37\,q^{42}+18\,q^{40}+20\,q^{38}+32\,q^{36}+9\,q^{34}+18\,q^{32
}+19\,q^{30}+6\,q^{28}+10\,q^{26}+12\,q^{24}+7\,q^{20}+4\,q^{18}+2\,q^
{14}+2\,q^{12}-q^{10}+q^{8} )$\\\\
eigenvalue: ${\frac {q^{13}}{2}}+q^{12}+{\frac {3\,q^{11}}{2}}+{\frac {5\,q^{10}}{2}}+{\frac {5\,q^{9}}{2}}+{\frac {5\,q^{8}}{2}}+{\frac {5\,q^{7}}{2}}+{\frac {3\,q^{6}}{2}}+{\frac {q^{5}}{2}}-{\frac {q^{3}}{2}}-q^{2}-1+{1 \over 2}\sqrt {a(q)}$\\
multiplicity: ${1 \over 2} (q^{57}+q^{56}+q^{55}+q^{54}+q^{53}+2\,q^{51}+2\,q^{50}+2\,q^{49}+q^{48
}+3\,q^{47}+q^{46}+4\,q^{45}+4\,q^{44}+4\,q^{43}+q^{42}+5\,q^{41}+2\,q
^{40}+6\,q^{39}+5\,q^{38}+6\,q^{37}+q^{36}+7\,q^{35}+4\,q^{34}+7\,q^{
33}+5\,q^{32}+7\,q^{31}+7\,q^{29}+5\,q^{28}+7\,q^{27}+4\,q^{26}+7\,q^{
25}+q^{24}+6\,q^{23}+5\,q^{22}+6\,q^{21}+2\,q^{20}+5\,q^{19}+q^{18}+4
\,q^{17}+4\,q^{16}+4\,q^{15}+q^{14}+3\,q^{13}+q^{12}+2\,q^{11}+2\,q^{
10}+2\,q^{9}+q^{7}+q^{6}+q^{5}+q^{4}+q^{3} )$\\\\
eigenvalue: ${\frac {q^{13}}{2}}+q^{12}+{\frac {3\,q^{11}}{2}}+{\frac {5\,q^{10}}{2}}+{\frac {5\,q^{9}}{2}}+{\frac {5\,q^{8}}{2}}+{\frac {5\,q^{7}}{2}}+{\frac {3\,q^{6}}{2}}+{\frac {q^{5}}{2}}-{\frac {q^{3}}{2}}-q^{2}-1-{1 \over 2}\sqrt {a(q)}$\\
multiplicity: ${ 1 \over 2} (q^{57}+q^{56}+q^{55}+q^{54}+q^{53}+2\,q^{51}+2\,q^{50}+2\,q^{49}+q^{48
}+3\,q^{47}+q^{46}+4\,q^{45}+4\,q^{44}+4\,q^{43}+q^{42}+5\,q^{41}+2\,q
^{40}+6\,q^{39}+5\,q^{38}+6\,q^{37}+q^{36}+7\,q^{35}+4\,q^{34}+7\,q^{
33}+5\,q^{32}+7\,q^{31}+7\,q^{29}+5\,q^{28}+7\,q^{27}+4\,q^{26}+7\,q^{
25}+q^{24}+6\,q^{23}+5\,q^{22}+6\,q^{21}+2\,q^{20}+5\,q^{19}+q^{18}+4
\,q^{17}+4\,q^{16}+4\,q^{15}+q^{14}+3\,q^{13}+q^{12}+2\,q^{11}+2\,q^{
10}+2\,q^{9}+q^{7}+q^{6}+q^{5}+q^{4}+q^{3} )$\\\\
eigenvalue: ${\frac {q^{11}}{2}}+{\frac {3\,q^{10}}{2}}+2\,q^{9}+{\frac {5\,q^{8}}{2}}+2\,q^{7}+{\frac {q^{6}}{2}}-{\frac {q^{5}}{2}}-q^{4}-q^{3}-q^{2}-1+{1 \over 2}\sqrt {b(q)}$\\
multiplicity: ${1 \over 2} (q^{78}+q^{77}+2\,q^{76}+q^{75}+2\,q^{74}+q^{73}+5\,q^{72}+3\,q^{71}+7
\,q^{70}+3\,q^{69}+8\,q^{68}+4\,q^{67}+14\,q^{66}+7\,q^{65}+17\,q^{64}
+7\,q^{63}+19\,q^{62}+9\,q^{61}+28\,q^{60}+13\,q^{59}+31\,q^{58}+12\,q
^{57}+33\,q^{56}+15\,q^{55}+44\,q^{54}+19\,q^{53}+45\,q^{52}+17\,q^{51
}+47\,q^{50}+21\,q^{49}+57\,q^{48}+23\,q^{47}+54\,q^{46}+20\,q^{45}+55
\,q^{44}+24\,q^{43}+62\,q^{42}+24\,q^{41}+55\,q^{40}+20\,q^{39}+54\,q^
{38}+23\,q^{37}+57\,q^{36}+21\,q^{35}+47\,q^{34}+17\,q^{33}+45\,q^{32}
+19\,q^{31}+44\,q^{30}+15\,q^{29}+33\,q^{28}+12\,q^{27}+31\,q^{26}+13
\,q^{25}+28\,q^{24}+9\,q^{23}+19\,q^{22}+7\,q^{21}+17\,q^{20}+7\,q^{19
}+14\,q^{18}+4\,q^{17}+8\,q^{16}+3\,q^{15}+7\,q^{14}+3\,q^{13}+5\,q^{
12}+q^{11}+2\,q^{10}+q^{9}+2\,q^{8}+q^{7}+q^{6} )$\\\\
eigenvalue: ${\frac {q^{11}}{2}}+{\frac {3\,q^{10}}{2}}+2\,q^{9}+{\frac {5\,q^{8}}{2}}+2\,q^{7}+{\frac {q^{6}}{2}}-{\frac {q^{5}}{2}}-q^{4}-q^{3}-q^{2}-1-{1 \over 2}\sqrt {b(q)}$\\
multiplicity: ${1 \over 2} (q^{78}+q^{77}+2\,q^{76}+q^{75}+2\,q^{74}+q^{73}+5\,q^{72}+3\,q^{71}+7
\,q^{70}+3\,q^{69}+8\,q^{68}+4\,q^{67}+14\,q^{66}+7\,q^{65}+17\,q^{64}
+7\,q^{63}+19\,q^{62}+9\,q^{61}+28\,q^{60}+13\,q^{59}+31\,q^{58}+12\,q
^{57}+33\,q^{56}+15\,q^{55}+44\,q^{54}+19\,q^{53}+45\,q^{52}+17\,q^{51
}+47\,q^{50}+21\,q^{49}+57\,q^{48}+23\,q^{47}+54\,q^{46}+20\,q^{45}+55
\,q^{44}+24\,q^{43}+62\,q^{42}+24\,q^{41}+55\,q^{40}+20\,q^{39}+54\,q^
{38}+23\,q^{37}+57\,q^{36}+21\,q^{35}+47\,q^{34}+17\,q^{33}+45\,q^{32}
+19\,q^{31}+44\,q^{30}+15\,q^{29}+33\,q^{28}+12\,q^{27}+31\,q^{26}+13
\,q^{25}+28\,q^{24}+9\,q^{23}+19\,q^{22}+7\,q^{21}+17\,q^{20}+7\,q^{19
}+14\,q^{18}+4\,q^{17}+8\,q^{16}+3\,q^{15}+7\,q^{14}+3\,q^{13}+5\,q^{
12}+q^{11}+2\,q^{10}+q^{9}+2\,q^{8}+q^{7}+q^{6} )$\\\\
eigenvalue: ${\frac {q^{13}}{2}}+q^{12}+{\frac {5\,q^{11}}{2}}+{\frac {7\,q^{10}}{2}}+{\frac {9\,q^{9}}{2}}+{\frac {9\,q^{8}}{2}}+{\frac {9\,q^{7}}{2}}+{\frac {5\,q^{6}}{2}}+{\frac {3\,q^{5}}{2}}-{\frac {q^{3}}{2}}-q^{2}-1+{1 \over 2}\sqrt {c(q)}$\\
multiplicity: ${1 \over 2} (q^{57}+q^{56}+q^{55}+q^{54}+q^{53}+2\,q^{52}+2\,q^{51}+2\,q^{50}+2\,q^
{49}+3\,q^{48}+3\,q^{47}+3\,q^{46}+4\,q^{45}+4\,q^{44}+4\,q^{43}+5\,q^
{42}+5\,q^{41}+6\,q^{40}+6\,q^{39}+5\,q^{38}+6\,q^{37}+7\,q^{36}+7\,q^
{35}+6\,q^{34}+7\,q^{33}+7\,q^{32}+7\,q^{31}+8\,q^{30}+7\,q^{29}+7\,q^
{28}+7\,q^{27}+6\,q^{26}+7\,q^{25}+7\,q^{24}+6\,q^{23}+5\,q^{22}+6\,q^
{21}+6\,q^{20}+5\,q^{19}+5\,q^{18}+4\,q^{17}+4\,q^{16}+4\,q^{15}+3\,q^
{14}+3\,q^{13}+3\,q^{12}+2\,q^{11}+2\,q^{10}+2\,q^{9}+2\,q^{8}+q^{7}+q
^{6}+q^{5}+q^{4}+q^{3} )$\\\\
eigenvalue: ${\frac {q^{13}}{2}}+q^{12}+{\frac {5\,q^{11}}{2}}+{\frac {7\,q^{10}}{2}}+{\frac {9\,q^{9}}{2}}+{\frac {9\,q^{8}}{2}}+{\frac {9\,q^{7}}{2}}+{\frac {5\,q^{6}}{2}}+{\frac {3\,q^{5}}{2}}-{\frac {q^{3}}{2}}-q^{2}-1-{1 \over 2}\sqrt {c(q)}$\\
multiplicity: ${1 \over 2} (q^{57}+q^{56}+q^{55}+q^{54}+q^{53}+2\,q^{52}+2\,q^{51}+2\,q^{50}+2\,q^
{49}+3\,q^{48}+3\,q^{47}+3\,q^{46}+4\,q^{45}+4\,q^{44}+4\,q^{43}+5\,q^
{42}+5\,q^{41}+6\,q^{40}+6\,q^{39}+5\,q^{38}+6\,q^{37}+7\,q^{36}+7\,q^
{35}+6\,q^{34}+7\,q^{33}+7\,q^{32}+7\,q^{31}+8\,q^{30}+7\,q^{29}+7\,q^
{28}+7\,q^{27}+6\,q^{26}+7\,q^{25}+7\,q^{24}+6\,q^{23}+5\,q^{22}+6\,q^
{21}+6\,q^{20}+5\,q^{19}+5\,q^{18}+4\,q^{17}+4\,q^{16}+4\,q^{15}+3\,q^
{14}+3\,q^{13}+3\,q^{12}+2\,q^{11}+2\,q^{10}+2\,q^{9}+2\,q^{8}+q^{7}+q
^{6}+q^{5}+q^{4}+q^{3} )$\\\\

\noindent
, where\\
       $a(q)= q^{26}+2\,q^{24}+2\,q^{23}+7\,q^{22}+12\,q^{21}+21\,q^{20}+30\,q^{19}+39\,q^{18}+46\,q^{17}+49\,q^{16}+46\,q^{15}+39\,q^{14}+30\,q^{13}+21\,q^{12}+12\,q^{11}+7\,q^{10}+2\,q^{9}+2\,q^{8}+q^{6}$\\
       $b(q)= q^{22}-2\,q^{21}+q^{20}+6\,q^{19}+18\,q^{18}+30\,q^{17}+37\,q^{16}+30\,q^{15}+18\,q^{14}+6\,q^{13}+q^{12}-2\,q^{11}+q^{10}$\\
       $c(q)= q^{26}-2\,q^{24}-2\,q^{23}-q^{22}+4\,q^{21}+17\,q^{20}+34\,q^{19}+51\,q^{18}+66\,q^{17}+73\,q^{16}+66\,q^{15}+51\,q^{14}+34\,q^{13}+17\,q^{12}+4\,q^{11}-q^{10}-2\,q^{9}-2\,q^{8}+q^{6}$\\
}

\section{$F_{4,1}$}
The graph $F_{4,1}$ is the 1-skeleton of the 24-cell on 24 vertices,
with spectrum $8^1 \z 4^4 \z 0^9 \z (-2)^8 \z (-4)^2$ (cf. \cite{BCN}, p. 314).\\\\

{\scriptsize

In the thick case $F_{4,1}(q)$ we find one of the metasymplectic spaces, described in \cite{Coh2}. The spectrum is:\\

\noindent
eigenvalue: $ {q}^{7}+{q}^{6}+{q}^{5}+2\,{q}^{4}+{q}^{3}+{q}^{2}+q$\\
multiplicity: $ 1$\\\\
eigenvalue: $ {q}^{5}+2\,{q}^{4}+{q}^{3}+{q}^{2}-1$\\
multiplicity: $ {\frac {{q}^{11}}{2}}+{q}^{8}+{\frac {{q}^{7}}{2}}+{\frac {{q}^{5}}{2}}+{q}^{4}+{\frac {q}{2}}$\\\\
eigenvalue: $ {q}^{4}-1$\\
multiplicity: $ {q}^{14}+{q}^{12}+2\,{q}^{10}+{q}^{8}+2\,{q}^{6}+{q}^{4}+{q}^{2}$\\\\
eigenvalue: $ -{q}^{3}-1$\\
multiplicity: $ {q}^{15}+{q}^{13}+{q}^{11}+2\,{q}^{9}+{q}^{7}+{q}^{5}+{q}^{3}$\\\\
eigenvalue: $ -{q}^{5}-{q}^{3}-{q}^{2}-1$\\
multiplicity: $ {\frac {{q}^{11}}{2}}+{\frac {{q}^{7}}{2}}+{\frac {{q}^{5}}{2}}+{\frac {q}{2}}$\\\\
}

\section{$H_{3,1}$ and $H_{3,3}$}
The graph $H_{3,1}$ is the 1-skeleton of the icosahedron, with spectrum
$$5^1 \z {\sqrt 5}^3 \z (-1)^5 \z (-{\sqrt 5})^3$$ (cf. \cite{CDS}, p. 310).

The graph $H_{3,3}$ is the 1-skeleton of the dodecahedron, with spectrum
$$3^1 \z {\sqrt 5}^3 \z 1^5 \z 0^4 \z (-2)^4 \z (-{\sqrt 5})^3$$
(cf. \cite{CDS}, p. 308).

\section{$H_{4,1}$ and $H_{4,4}$}
The graph $H_{4,1}$ is the 1-skeleton of the 600-cell on 120 vertices.
It has spectrum
$$12^1 \z (3+3{\sqrt 5})^4 \z (2+2{\sqrt 5})^9 \z 3^{16} \z 0^{25} ~
(-2)^{36} \z (2-2{\sqrt 5})^9 \z (-3)^{16} \z (3-3{\sqrt 5})^4.$$
Note that the vertices of $H_{4,1}$ may be identified with the elements
of the group $G = SL(2,5)$ (cf. \cite{BCN}, p. 315), and the graph is
invariant under right multiplication by elements of $G$, so that
we have the right regular representation of $G$.
This explains why the multiplicities are the squares
of the degrees of the irreducible characters of $SL(2,5)$.

The graph $H_{4,4}$ is the 1-skeleton of the 120-cell on 600 vertices.
(The vertices can be regarded as 4-cliques in $H_{4,1}$, adjacent
when they have a triangle in common.)
It has spectrum

\noindent
\begin{tabular}{ccc}
approx. ev.&eigenvalue&mult. \\
4.000000&4&1 \\
3.854102&$3 \tau - 1$&4 \\
3.618034&$\tau + 2$&9 \\
3.302776&${1 \over 2}(3+{\sqrt {13}})$&16 \\
2.925423&$\gamma$&25 \\
2.518199&$\zeta$&36 \\
2.236068&${\sqrt 5}$&24 \\
1.791288&${1 \over 2}(-1+{\sqrt {21}})$&16 \\
1.618034&$\tau$&24 \\
1.381966&$3 - \tau$&9 \\
1.178194&$\eta$&36 \\
1.000000&1&40 \\
0.551929&$\delta$&25 \\
0.414214&${\sqrt 2} - 1$&48
\end{tabular}
\begin{tabular}{ccc}
approx. ev.&eigenvalue&mult. \\
0.000000&0&18 \\
$-0.302776$&${1 \over 2}(3-{\sqrt {13}})$&16 \\
$-0.381966$&$\tau - 2$&30 \\
$-0.618034$&$1 - \tau$&24 \\
$-1.000000$&$-1$&8 \\
$-2.000000$&$-2$&8 \\
$-2.236068$&$-{\sqrt 5}$&24 \\
$-2.414214$&$-{\sqrt 2} - 1$&48 \\
$-2.477352$&$\epsilon$&25 \\
$-2.618034$&$-1 - \tau$&30 \\
$-2.696393$&$\theta$&36 \\
$-2.791288$&${1 \over 2}(-1-{\sqrt {21}})$&16 \\
$-2.854102$&$2 - 3 \tau$&4
\end{tabular}

Here $\gamma$, $\delta$, $\epsilon$ are the three roots of $x^3 - x^2 - 7x + 4 = 0$,
and $\zeta$, $\eta$, $\theta$ are the three roots of $x^3 - x^2 -7x + 8 = 0$,
and $\tau = (1 + {\sqrt 5})/2$.

In this case the association scheme has $d+1 = 45$ relations, while
$L_1$ has only 27 distinct eigenvalues. Indeed, every eigenvalue
of $L_1$ that has degree $m$ over $\Qq(\tau)$ occurs in $L_1$ with
multiplicity $m$.

\section{$I^m_{2,1}$}
The graph $I^m_{2,1}$ is the $m$-gon, with spectrum $2 \cos (2 \pi j / m)$
$(0 \le j \le m-1)$ (cf. \cite{CDS}, p. 53).\\\\

{\scriptsize

In the thick case we only have to look at $m=6$ or $m=8$. 

In the case $I^6_{2,1}(q) = G_{2,1}(q)$ we find a generalized hexagon which is distance-regular with spectrum:

\noindent
eigenvalue: $ {q}^{2}+q$\\
multiplicity: $ 1 $\\\\
eigenvalue: $ -1+2\,q$\\
multiplicity: $ {\frac {{q}^{5}}{6}}+{\frac {{q}^{4}}{2}}+{\frac {2\,{q}^{3}}{3}}+{\frac {{q}^{2}}{2}}+{\frac {q}{6}} $\\\\
eigenvalue: $ -q-1$\\
multiplicity: $ {\frac {{q}^{5}}{3}}+{\frac {{q}^{3}}{3}}+{\frac {q}{3}}$\\\\
eigenvalue: $-1$\\
multiplicity: $ {\frac {{q}^{5}}{2}}+{\frac {{q}^{4}}{2}}+{\frac {{q}^{2}}{2}}+{\frac{q}{2}} $\\\\

In the case $I^8_{2,1}(q)$ with $q=2^{k+\frac{1}{2}}$ we find a generalized octagon which is distance-regular with spectrum:

\noindent
eigenvalue: $ q^6+q^2$\\
multiplicity: $ 1 $\\\\
eigenvalue: $ {q}^{2}-1 $\\
multiplicity: $ {\frac {{q}^{20}}{2}}+{\frac {{q}^{16}}{2}}+{\frac {{q}^{8}}{2}}+{\frac {{q}^{4}}{2}} $\\\\
eigenvalue: $ -{q}^{4}-1 $\\
multiplicity: $ {q}^{14}-{q}^{12}+{q}^{8}-{q}^{4}+{q}^{2} $\\\\
eigenvalue: $ -1+{q}^{2}-\sqrt {2}{q}^{3} $\\
multiplicity: $ 2^{4k}+2^{5k+1}+2^{6k+2}-2^{8k+2}-2^{9k+4}+2^{11k+4}+2^{12k+6}+2^{13k+5}-2^{15k+7}-2^{16k+6}+2^{18k+8}+2^{19k+8}+2^{20k+8} $\\\\
eigenvalue: $ -1+{q}^{2}+\sqrt {2}{q}^{3} $\\
multiplicity: $ 2^{4k}-2^{5k+1}+2^{6k+2}-2^{8k+2}+2^{9k+4}-2^{11k+4}+2^{12k+6}-2^{13k+5}+2^{15k+7}-2^{16k+6}+2^{18k+8}-2^{19k+8}+2^{20k+8} $\\\\
}

\section{Remark}
This research has been done in 1993. We found that Yasushi Gomi (\cite{Gom1}
and \cite{Gom2}) computed some of the results mentioned here in the context
of computing character tables of (commutative) Hecke algebras.
His results agree with ours.








\acknowledgements
We thank Henny Wilbrink for useful discussions.
In the thick case the computations were done in Maple.

\begin{references}
\bibitem{BC} A.E. Brouwer \& A.M. Cohen,
`` Computation of some parameters of Lie geometries '',
Annals of Discr. Math. 26 (1983) 1-48

\bibitem{BCN} A.E. Brouwer, A.M. Cohen \& A. Neumaier,
`` Distance-regular graphs '',
Springer, Heidelberg, 1989

\bibitem{Coh2}A.M. Cohen, `` Points and Lines in Metasymplectic Spaces '',
Annals of Discrete Mathematics 18, 1983, pp. 193-196

\bibitem{CDS} D.M. Cvetkovi\'c, M. Doob \& H. Sachs,
`` Spectra of graphs '',
V.E.B. Deutscher Verlag der Wissenschaften, Berlin, 1979 
and Academic Press, New York, 1980

\bibitem{Gom1}Yasushi Gomi,
`` Character tables of commutative Hecke algebras associated with 
exceptional Weyl groups'',
Comm. in Alg., 22(1), 1994, pp. 123-138

\bibitem{Gom2}Yasushi Gomi, 
`` Character tables of commutative Hecke algebras associated with finite Chevalley groups of exceptional type '',
Comm. in Alg., 22(11), 1994, pp. 4361-4372

\end{references}

\end{article}

\end{document}
