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\title{Classification of the (0,2)-graphs of valency 8}
\author{Andries E. Brouwer \& Patric R. J. \"Osterg{\aa}rd}
% \date{March 15, 2006}
\date{December 14, 2006}

\begin{document}
\maketitle
\hyphenation{semi-biplane}

\begin{abstract}
We find the 302 graphs of valency 8 with the property that
any two nonadjacent vertices have 0 or 2 common neighbours,
and, in particular, the 104 semibiplanes of block size 8.
\end{abstract}

\section{Introduction}
A {\em $(0,2)$-graph} is a connected graph such that
any two vertices have either 0 or 2 common neighbours.
A {\em rectagraph} is a $(0,2)$-graph without triangles.
A {\em semibiplane} is a connected point-block incidence
structure such that any two points are in either 0 or 2
common blocks, and any two blocks have either 0 or 2
common points. Clearly, the incidence graph of a semibiplane
is a bipartite $(0,2)$-graph (and conversely).

Martyn Mulder \cite{Mu} showed that $(0,2)$-graphs are regular,
say of valency $k$, and that the number $v$ of vertices
and the diameter $d$ satisfy $v \le 2^k$ and $d \le k$
with in both cases equality only for the hypercube.

Peter Wild \cite{W} determined all semi\-biplanes of block size at most 6.
Michel Mollard \cite{Mo} determined all $(0,2)$-graphs on
at most 31 vertices (and finds that these have valency at most 7).
Andries Brouwer \cite{B} determined all $(0,2)$-graphs with
valency at most 7.

In this note we find all $(0,2)$-graphs with valency 8.
It turns out that the total number $N(k)$ of nonisomorphic
$(0,2)$-graphs with $k \le 8$ is given by

\medskip\begin{center}
\begin{tabular}{|c|rrrrrrrrr|}
\hline
$k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
$N(k)$ & 1 & 1 & 1 & 2 & 3 & 8 & 24 & 96 & 302 \\
\hline
\end{tabular}
\end{center}

\section{Classification algorithms}
\subsection*{Reduction to the bipartite case}
We first classify the bipartite $(0,2)$-graphs with given valency $k$.
Given that classification, the nonbipartite $(0,2)$-graphs of that valency
are easily found by observing that any nonbipartite $(0,2)$-graph $\Gamma$
produces a bipartite $(0,2)$-graph $\Gamma'$ with the same valency
when the following doubling construction is applied: For each vertex $x$
in $\Gamma$, there are two vertices $x^{+}, x^{-}$ in $\Gamma'$
and for each edge $xy$ in $\Gamma$, there are two edges 
$x^{+}y^{-}, x^{-}y^{+}$ in $\Gamma'$. This graph $\Gamma'$
(known as the {\em bipartite double} of $\Gamma$) has an
automorphism $\sigma$ of order two without fixed edges
interchanging $x^{+}$ and $x^{-}$. And conversely, given any
bipartite $(0,2)$-graph $\Delta$ with an automorphism of order 2
without fixed edges that interchanges the two classes of the bipartition,
the quotient $\Delta / \sigma$ will be a nonbipartite $(0,2)$-graph.
In this way the entire classification is reduced to the bipartite case.

\subsection*{Classification in the bipartite case}
In the classification, 
we first fix one vertex, $u$, and then for increasing
$d$ build up graphs with vertices at distance
at most $d$ from $u$ that fulfill some
necessary conditions for being an induced subgraph of
a bipartite (0,2)-graph with given valency $k$.

Let $S_i = \{ x : d(x,u) = i\}$. Clearly $|S_1|=k$,
and we let $S_1 = \{1,2,\ldots ,k\}$. For $i > 1$, we 
may denote any element $v$ of $S_i$ by 
$\{w : w \in S_{i-1} \cap N(v)\}$, where $N(v)$
is the neighborhood of $v$. (This works, since 
$S_{i-1} \cap N(v) \neq S_{i-1} \cap N(v')$ for
any distinct $v,v' \in S_i$. Indeed, this is clear for $i = 2$,
and follows from $|S_{i-1} \cap N(v)| \ge i$ for $i > 2$.)

Since every vertex in $S_2$ has at least one common
neighbor with $u$, it must have exactly two
neighbors in $S_1$. Moreover, since every two distinct 
vertices in $S_1$ have $u$ as a neighbor, they must have 
exactly one neighbor in $S_2$. 
Consequently, $S_2 = \{\{i,j\} : 1 \leq i < j \leq k$\}, 
and we may view the elements of $S_2$ as the edges of a
complete graph of order $k$.

We now consider the case where, for some $i \geq 2$, we
have built up a graph with vertices at distance at most
$i$ from $u$, and try to find the vertices in $S_{i+1}$.
Such an extension consists of two steps: finding candidates
for vertices in $S_{i+1}$---more precisely, we determine
possible neighborhoods in $S_i$ for such a vertex---and
finding a subset of candidates
that form $S_{i+1}$. We will now have a closer
look at these two steps. Throughout the search, necessary
conditions for the complete graph to be a $(0,2)$-graph
are used for pruning.

In finding candidates for vertices in $S_{i+1}$ we focus
on the fact that a vertex in $S_{i+1}$ and a vertex in
$S_{i-1}$ should have either 0 or 2 common neighbors in 
$S_i$. We therefore say that a set of vertices 
$v_1,v_2,\ldots ,v_m \in S_i$ forms a candidate 
if $\cup_{i=1}^m v_i$ as a multiset contains each element
of $S_{i-1}$ 0 or 2 times and $m \leq k$. 
Such candidates can be found
by backtrack search: A partial candidate should cover
the elements of $S_{i-1}$ 0, 1, or 2 times; one may prune
the search if the number of elements covered once exceeds
the number of distinct elements that can be covered by
the vertices in $S_i$ still to be added (utilizing the
fact that $m \leq k$ and knowing the maximum size of
any set $v_i$).

Determining the candidates for vertices in $S_3$, where
the actual computer search starts, has a nice interpretation
in terms of graphs of order $k$ (when the vertices in
$S_2$ are interpreted as the edges in the complete graph
of order $k$). Namely, the candidates then correspond
to the 2-regular graphs with vertices from 
$\{1,2,\ldots ,k\}$. Such graphs clearly consist of
sets of cycles and their isomorphism classes can be
characterized by the length of the cycles. For $k=8$,
considered in this study, they are 
\begin{equation}
\label{types}
4+4,\quad 5+3,\quad 8,\quad
4+3,\quad 7,\quad 3+3,\quad 6,\quad 5,\quad 4,\quad 3.
\end{equation}
(From now on, we occasionally consider the specific
case $k=8$ for clarity. However, all ideas can be
applied for arbitrary $k$.)

In searching for subsets of candidates that 
form $S_{i+1}$ we focus on the fact that two distinct
vertices in $S_i$ should have either 0 or 2 common
neighbors. Now we know their common neighbors
in $S_{i-1}$; if this number is 0, 1, or 2, then the
pair of vertices should occur in the neighborhood 
of 0 or 2, 1, or
0 vertices in $S_{i+1}$, respectively. In the last of
these three cases, we can immediately reject 
corresponding candidates. It is not straightforward to
implement the requirement in the first of the cases in
a computer search, so the core of the algorithm
is to search for candidates that cover the pairs of vertices 
in $S_i$ with exactly one common neighbor in $S_{i-1}$   
exactly once. In fact, we then have instances of the
\emph{exact cover} problem \cite{K}.

Checking the requirement that two vertices of $S_i$ that do
not have common neighbors in $S_{i-1}$ must have 0
or 2 common neighbors in $S_{i+1}$ cannot---for efficiency
reasons---be postponed until the solutions of the 
aforementioned exact cover problem are found. 
Terming a vertex 
\emph{saturated} if its valency is $k$, whenever
a vertex $v \in S_i$ gets saturated when building up
$S_{i+1}$ with the exact cover algorithm, we carry out
this pruning test for all pairs in $S_i$ where $v$ is one 
of the vertices.

All solutions of the exact cover problem with the 
additional pruning test are immediately subjected to one
more test: two vertices in $S_{i+1}$ cannot have more
than 2 common neighbors in $S_i$. 

To make the outlined algorithm efficient enough for 
settling the case $k=8$ in reasonable time, we need
to incorporate isomorph rejection in some form. This
is done during the search for $S_3$, which is the
hardest part of this classification problem. The search
for $S_3$ is divided into ten parts, corresponding to
the cycle types (\ref{types}). Numbering these types
from 1 to 10, in part $i \in \{1,2,\ldots ,10\}$  
of the search the first vertex
has type $i$ (there is a unique choice, up to isomorphism)
and the type of the other vertices must be greater
than or equal to $i$. The ordering of the types is of
great importance; the ordering in (\ref{types}), in
increasing order with respect to the value of $i$, works 
reasonably well.

For each of the aforementioned ten parts, $S_3$ is
constructed in two steps. Consider a neighbor $v$ in $S_2$
of the vertex that is fixed first (the choice of $v$ will
occasionally have some---but not a crucial---impact on the 
performance). Then we first determine all vertices in $S_3$ 
that have $v$ as a neighbor. During and after this search
partial solutions are also subjected to isomorphism tests,
using \emph{nauty} \cite{Mc}. From there on, we search for 
the rest of $S_3$ with the approach outlined earlier.

Solutions for $S_i$ with $i \geq 4$ can be obtained 
directly by the described algorithm. Finally, isomorph
rejection has to be carried out among the complete 
$(0,2)$-graphs.

\subsection*{Classification results}

Whereas the classification of $(0,2)$-graphs with
valency 7 using the outlined algorithm 
takes only minutes of CPU time on a contemporary
personal computer, the classification
for valency 8 takes about one month. Below we
present the main parameters of the graphs obtained and 
discuss some of their properties.

Actually, not all of the classification was done by computer.
Some parts of extending (the unique) $S_3$ 
obtained in case 10 to a complete $(0,2)$-graph turned 
out to be computationally challenging. Fortunately, however,
this is precisely the case that is well-understood. We have
a rectagraph of valency $k$ such that any 3-claw determines
a unique 3-cube. By \cite[4.3.6, 4.3.8]{BCN}, the graph is a
quotient of a hypercube. That means that the vertices of the
graph can be labeled with binary words of length 8, where
the set of words at each vertex has even minimum distance larger
than 6, that is, 8. Up to equivalence there are two such codes,
the trivial one-word code and the repetition code,
and consequently two such graphs (since twists cannot occur),
the 8-cube on 256 vertices, and its antipodal quotient on 128 vertices.

To get confidence in the validity of these 
computational results, we carried out 
double counting tests described
in \cite[Sect.\ 10.3]{KO}. Specifically, for each graph
obtained, we determined its induced subgraphs consisting of
one vertex and the vertices at distance at most 3,
and calculated analytically the number of times
the complete graph should have been produced
(this is possible since from $S_3$ onwards, no isomorph
rejection is carried out before the graph is completed).
We have to take into account though the missing
computational results in case 10. 
Similar double counting was also carried out
in the construction of $S_3$.

\section{Semibiplanes with $k = 8$}

The number $N(k)$ of nonisomorphic bipartite $(0,2)$-graphs with $k \le 8$
is given in the following table.

\medskip\begin{center}
\begin{tabular}{|c|rrrrrrrrr|}
\hline
$k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
$N(k)$ & 1 & 1 & 1 & 1 & 2 & 4 & 13 & 40 & 104 \\
\hline
\end{tabular}
\end{center}

\bigskip
In the table below, \# gives a serial number (given $k$),
$d$ denotes the diameter, `bipd' stands for bipartite double,
`ebipd' for extended bipartite double,
`drg' means distance-regular graph, `ico' means icosahedron.
The `distribution' column gives the distance distribution:
the number of points at each distance from a given point.
If the group is nontransitive, there may be several different
distributions, depending on the choice of the given point.
The `gsz' column gives the size of the automorphism group $G$.

The 'graph' column may give a construction or comment.
In the case of quotients $2^8/C$ the code $C$ is given by generators;
a vector like $11100000$ is abbreviated $1^30^5$.

All semibiplanes with block size $k$ are self-dual for $1 \le k \le 7$,
but for $k = 8$ there are 15 non-self-dual examples, and non-selfduality
is noted in the comments.

The bipartite $(0,2)$-graph of valency $k$ with serial number $i$
is called $\Delta_{k.i}$. Serial numbers for $k \le 7$ are as in \cite{B}.

Graph $\Delta_{8.2}$ is the quotient $\Pi/\sigma$, where $\Pi$ is the
projective plane $PG(2,8)$ and $\sigma$ an elation, and the quotient
is the semibiplane consisting of the $\sigma$-orbits of size 2
on points and lines. It is a distance-regular graph
with parameters drg(8,7,6,1; 1,2,7,8), cf. \cite[p. 224]{BCN}.

In some cases the second largest eigenvalue $\lambda_2$ is given,
in order to distinguish parameter sets that would otherwise have
identical data.

\medskip{\footnotesize
\tablehead{
\# & $v$ & $d$ & distribution & gsz & orbits & graph \\
\hline}
\tabletail{\hline}
\noindent\begin{supertabular}{|r|rrlrll}
1 & 64 & 4 & 1+8+28+24+3 & 384 & tra & \\ % #0010
2 & 64 & 4 & 1+8+28+24+3 & 10752 & tra & $\Pi/\sigma$, drg \\ % #0061
3 & 68 & 4 & 1+8+28+26+5 & 4 & 17*4 & $\lambda_2 = 3.708$ \\ % #0001
4 & 68 & 4 & 1+8+28+26+5 & 20 & 2*4+3*20 & $\lambda_2 = 3.689$ \\ % #0002
5 & 68 & 4 & 1+8+28+26+5 & 4 & 17*4 & $\lambda_2 = 3.755$ \\ % #0003
6 & 68 & 4 & 1+8+28+26+5 & 20 & 2*4+3*20 & $\lambda_2 = 3.742$ \\ % #0004
7 & 72 & 4 & 1+8+28+28+7 & 144 & tra & \\ % #0103
8 & 74 & 5 & 1+8+28+28+8+1 (18x) & 672 & 2+16+56 & \\ % #0071
  &    &   & 1+8+28+29+8 (56x) \\
9 & 80 & 4 & 1+8+28+32+11 & 4 & 8*2+16*4 & not selfdual\\ % #0034
10 & 80 & 4 & 1+8+28+32+11 & 20 & 4*20 & not selfdual\\ % #0036
11 & 80 & 4 & 1+8+28+32+11 & 20 & 4*20 & \\ % #0037
12 & 80 & 4 & 1+8+28+32+11 & 384 & 32+48 & \\ % #0060
13 & 80 & 5 & 1+8+28+31+11+1 & 3840 & tra & cf. \S\ref{aq}, \ref{gq} \\ % s8.104
14 & 80 & 5 & 1+8+28+31+11+1 (8x) & 12 & 2*4+6*12 & not selfdual\\ % #0045
   &    &   & 1+8+28+32+11 (72x) \\
15 & 80 & 5 & 1+8+28+31+11+1 (16x) & 768 & 16+64 & \\ % #0059
   &    &   & 1+8+28+32+11 (64x) \\
16 & 84 & 4 & 1+8+28+34+13 & 16 & 4+6*8+2*16 & \\ % #0016
17 & 84 & 4 & 1+8+28+34+13 & 64 & 4+16+2*32 & \\ % #0056
18 & 84 & 4 & 1+8+28+34+13 & 32 & 4+2*8+2*16+32 & \\ % #0062
19 & 84 & 4 & 1+8+28+34+13 & 1344 & tra & cf. \S\ref{dircov} \\ % #0081
20 & 84 & 5 & 1+8+28+33+13+1 (8x) & 8 & 7*4+7*8 & $\lambda_2 = 4.243$\\ % #0005
   &    &   & 1+8+28+34+13 (76x) \\
21 & 84 & 5 & 1+8+28+33+13+1 (8x) & 8 & 7*4+7*8 & $\lambda_2 = 4.321$\\ % #0006
   &    &   & 1+8+28+34+13 (76x) \\
22 & 84 & 5 & 1+8+28+33+13+1 (8x) & 4 & 14*2+14*4 & not selfdual\\ % #0007
   &    &   & 1+8+28+34+13 (76x) \\
23 & 84 & 5 & 1+8+28+33+13+1 (4x) & 4 & 14*2+14*4 & not selfdual\\ % #0008
   &    &   & 1+8+28+34+13 (80x) \\
24 & 84 & 5 & 1+8+28+33+13+1 (20x) & 16 & 4+6*8+2*16 & \\ % #0009
   &    &   & 1+8+28+34+13 (64x) \\
25 & 84 & 5 & 1+8+28+33+13+1 (28x) & 16 & 3*4+3*8+3*16 & \\ % #0017
   &    &   & 1+8+28+34+13 (56x) \\
26 & 84 & 5 & 1+8+28+33+13+1 (52x) & 64 & 4+16+2*32 & \\ % #0020
   &    &   & 1+8+28+34+13 (32x) \\
27 & 84 & 5 & 1+8+28+33+13+1 (8x) & 16 & 3*4+3*8+3*16 & \\ % #0038
   &    &   & 1+8+28+34+13 (76x) \\
28 & 84 & 5 & 1+8+28+33+13+1 (20x) & 64 & 4+16+2*32 & \\ % #0055
   &    &   & 1+8+28+34+13 (64x) \\
29 & 84 & 5 & 1+8+28+33+13+1 (36x) & 96 & 12+24+48 & \\ % #0057
   &    &   & 1+8+28+34+13 (48x) \\
30 & 88 & 4 & 1+8+28+36+15 & 160 & 8+40+4 & \\ % #0023
31 & 88 & 5 & 1+8+28+35+15+1 (16x) & 16 & 3*8+4*16 & \\ % #0013
   &    &   & 1+8+28+36+15 (72x) \\
32 & 88 & 5 & 1+8+28+35+15+1 (32x) & 32 & 3*8+2*32 & \\ % #0033
   &    &   & 1+8+28+36+15 (56x) \\
33 & 96 & 4 & 1+8+28+40+19 & 1152 & tra & \\ % #0049
34 & 96 & 5 & 1+8+28+37+19+3 & 5760 & tra & ebipd($\Gamma_{7.44}$)\\ % #0018
35 & 96 & 5 & 1+8+28+37+19+3 (64x) & 384 & 32+64 & \\ % #0019
   &    &   & 1+8+28+40+19 (32x) \\
36 & 96 & 5 & 1+8+28+39+19+1 & 24 & 2*12+3*24 & \\ % #0021
37 & 96 & 5 & 1+8+28+38+19+2 & 768 & tra & $2\times\Delta_{7.1}$ \\ % #0024
38 & 96 & 5 & 1+8+28+38+19+2 & 3840 & tra & $2\times\Delta_{7.2}$, ebipd($\Gamma_{7.45}$) \\ % #0025
39 & 96 & 5 & 1+8+28+38+19+2 & 192 & tra & $2\times\Delta_{7.9}$ \\ % #0026
40 & 96 & 5 & 1+8+28+38+19+2 & 128 & 32+64 & $2\times\Delta_{7.4}$ \\ % #0027
41 & 96 & 5 & 1+8+28+38+19+2 & 128 & 32+64 & $2\times\Delta_{7.5}$ \\ % #0028
42 & 96 & 5 & 1+8+28+38+19+2 & 96 & tra & $2\times\Delta_{7.6}$ \\ % #0029
43 & 96 & 5 & 1+8+28+38+19+2 & 144 & 24+72 & $2\times\Delta_{7.8}$ \\ % #0030
44 & 96 & 5 & 1+8+28+38+19+2 & 192 & tra & $2\times\Delta_{7.7}$ \\ % #0031
45 & 96 & 5 & 1+8+28+38+19+2 & 96 & tra & $2\times\Delta_{7.3}$ \\ % #0032
46 & 96 & 5 & 1+8+28+38+19+2 (64x) & 128 & 32+64 & \\ % #0035
   &    &   & 1+8+28+40+19 (32x) \\
47 & 96 & 5 & 1+8+28+38+19+2 (12x) & 12 & 12*6+2*12 & not selfdual\\ % #0050
   &    &   & 1+8+28+39+19+1 (48x) \\
   &    &   & 1+8+28+40+19 (36x) \\
48 & 96 & 5 & 1+8+28+38+19+2 (12x) & 24 & 8*12 & not selfdual\\ % #0051
   &    &   & 1+8+28+39+19+1 (48x) \\
   &    &   & 1+8+28+40+19 (36x) \\
49 & 96 & 5 & 1+8+28+39+19+1 (48x) & 48 & 4*24 & not selfdual\\ % #0052
   &    &   & 1+8+28+40+19 (48x) \\
50 & 96 & 5 & 1+8+28+39+19+1 & 96 & 48+48 & \\ % #0054
51 & 96 & 5 & 1+8+28+39+19+1 & 72 & 24+36+36 & \\ % #0058
52 & 96 & 5 & 1+8+28+38+19+2 & 4032 & tra & $2\times\Delta_{7.10}$ \\ % #0066
53 & 100 & 5 & 1+8+28+39+21+3 (24x) & 2 & 28*1+36*2 & not selfdual\\ % #0011
   &     &   & 1+8+28+40+21+2 (46x) \\
   &     &   & 1+8+28+41+21+1 (26x) \\
   &     &   & 1+8+28+42+21 (4x) \\
54 & 100 & 5 & 1+8+28+39+21+3 (16x) & 4 & 25*4 & \\ % #0014
   &     &   & 1+8+28+40+21+2 (56x) \\
   &     &   & 1+8+28+41+21+1 (24x) \\
   &     &   & 1+8+28+42+21 (4x) \\
55 & 100 & 5 & 1+8+28+38+21+4 (2x) & 2 & 22*1+39*2 & not selfdual\\ % #0015
   &     &   & 1+8+28+39+21+3 (4x) \\
   &     &   & 1+8+28+40+21+2 (32x) \\
   &     &   & 1+8+28+41+21+1 (54x) \\
   &     &   & 1+8+28+42+21 (8x) \\
56 & 100 & 5 & 1+8+28+39+21+3 (36x) & 8 & 4*2+5*4+9*8 & \\ % #0022
   &     &   & 1+8+28+40+21+2 (40x) \\
   &     &   & 1+8+28+41+21+1 (14x) \\
   &     &   & 1+8+28+42+21 (10x) \\
57 & 100 & 5 & 1+8+28+40+21+2 (40x) & 20 & 5*20 & \\ % #0065
   &     &   & 1+8+28+41+21+1 (40x) \\
   &     &   & 1+8+28+42+21 (20x) \\
58 & 104 & 5 & 1+8+28+38+23+6 (8x) & 32 & 3*8+3*16+32 & \\ % #0012
   &     &   & 1+8+28+39+23+5 (8x) \\
   &     &   & 1+8+28+40+23+4 (32x) \\
   &     &   & 1+8+28+41+23+3 (16x) \\
   &     &   & 1+8+28+42+23+2 (40x) \\
59 & 104 & 5 & 1+8+28+40+23+4 (8x) & 48 & 8+4*24 & \\ % #0072
   &     &   & 1+8+28+42+23+2 (96x) \\
60 & 104 & 5 & 1+8+28+38+23+6 (12x) & 96 & 4+12+16+24+48 & \\ % #0073
   &     &   & 1+8+28+40+23+4 (52x) \\
   &     &   & 1+8+28+42+23+2 (24x) \\
   &     &   & 1+8+28+43+23+1 (16x) \\
61 & 104 & 5 & 1+8+28+40+23+4 (64x) & 1536 & 16+24+64 & \\ % #0083
   &     &   & 1+8+28+42+23+2 (24x) \\
   &     &   & 1+8+28+44+23 (16x) \\
62 & 112 & 4 & 1+8+28+48+27 & 21504 & tra & cf. \S\ref{l32} \\ % #0099
63 & 112 & 5 & 1+8+28+42+27+6 & 16 & 4*8+5*16 & $2\times\Delta_{7.12}$ \\ % #0039
64 & 112 & 5 & 1+8+28+42+27+6 & 16 & 4*8+5*16 & $2\times\Delta_{7.15}$ \\ % #0041
65 & 112 & 5 & 1+8+28+42+27+6 & 48 & 4+12+4*24 & $2\times\Delta_{7.14}$ \\ % #0042
66 & 112 & 5 & 1+8+28+42+27+6 & 32 & 3*16+2*32 & $2\times\Delta_{7.17}$ \\ % #0044
67 & 112 & 5 & 1+8+28+42+27+6 & 32 & 2*8+2*16+2*32 & $2\times\Delta_{7.11}$ \\ % #0046
68 & 112 & 5 & 1+8+28+42+27+6 & 32 & 2*8+2*16+2*32 & $2\times\Delta_{7.13}$ \\ % #0047
69 & 112 & 5 & 1+8+28+42+27+6 & 8 & 14*8 & $2\times\Delta_{7.16}$ \\ % #0053
70 & 116 & 6 & 1+8+28+42+28+8+1 (4x) & 672 & 4+28+84 & \\ % #0097
   &     &   & 1+8+28+46+29+4 (112x) \\
71 & 128 & 4 & 1+8+28+56+35 & 5160960 & tra & $2^8/{\bf 1}$, ebipd($\Gamma_{7.51}$)\\ % #0101
72 & 128 & 5 & 1+8+28+46+35+10 & 12288 & tra & $2^2\times\Delta_{6.1}$ \\ % #0067
73 & 128 & 5 & 1+8+28+46+35+10 & 6144 & tra & $2^2\times\Delta_{6.2}$ \\ % #0068
74 & 128 & 5 & 1+8+28+46+35+10 & 184320 & tra & $2^2\times\Delta_{6.3}$, ebipd($\Gamma_{7.52}$) \\ % #0069
75 & 128 & 5 & 1+8+28+46+35+10 (64x) & 92160 & 64+64 & $2^8/\pi_{(12)}(00111111)$, \\ % #0070
   &     &   & 1+8+28+56+35 (64x) & & & ebipd($\Gamma_{7.53}$) \\
76 & 128 & 5 & 1+8+28+46+35+10 & 96 & 32+96 & $2\times\Delta_{7.18}$ \\ % #0074
77 & 128 & 5 & 1+8+28+46+35+10 & 2304 & tra & $2\times\Delta_{7.22}$ \\ % #0077
78 & 128 & 5 & 1+8+28+46+35+10 & 240 & 48+80 & $2\times\Delta_{7.19}$ \\ % #0082
79 & 128 & 6 & 1+8+28+45+34+11+1 (48x) & 16 & 8*8+4*16 & $2\times\Delta_{7.26}$ \\ % #0040
   &     &   & 1+8+28+46+35+10 (80x) \\
80 & 128 & 6 & 1+8+28+45+34+11+1 (72x) & 48 & 8+5*24 & $2\times\Delta_{7.25}$ \\ % #0043
   &     &   & 1+8+28+46+35+10 (56x) \\
81 & 128 & 6 & 1+8+28+45+34+11+1 (8x) & 24 & 8+5*24 & $2\times\Delta_{7.24}$ \\ % #0048
   &     &   & 1+8+28+46+35+10 (120x) \\
82 & 128 & 6 & 1+8+28+45+34+11+1 (28x) & 24 & 2*4+2*12+4*24 & $2\times\Delta_{7.27}$ \\ % #0063
   &     &   & 1+8+28+46+35+10 (100x) \\
83 & 128 & 6 & 1+8+28+45+34+11+1 (16x) & 672 & 16+112 & $2\times\Delta_{7.28}$ \\ % #0064
   &     &   & 1+8+28+46+35+10 (112x) \\
84 & 128 & 6 & 1+8+28+48+34+8+1 (24x) & 384 & 16+2*24+64 & not selfdual\\ % #0093
   &     &   & 1+8+28+48+35+8 (24x) \\
   &     &   & 1+8+28+49+35+7 (64x) \\
   &     &   & 1+8+28+52+35+4 (16x) \\
85 & 128 & 6 & 1+8+28+48+34+8+1 (2x) & 48 & 2+6+8+16+2*48 & not selfdual\\ % #0094
   &     &   & 1+8+28+48+35+8 (6x) \\
   &     &   & 1+8+28+49+35+7 (112x) \\
   &     &   & 1+8+28+50+35+6 (8x) \\
86 & 128 & 6 & 1+8+28+48+34+8+1 (8x) & 64 & 2*8+16+32+64 & not selfdual\\ % #0095
   &     &   & 1+8+28+48+35+8 (24x) \\
   &     &   & 1+8+28+49+35+7 (64x) \\
   &     &   & 1+8+28+50+35+6 (32x) \\
87 & 128 & 6 & 1+8+28+48+34+8+1 (16x) & 512 & 2*16+32+64 & not selfdual\\ % #0096
   &     &   & 1+8+28+48+35+8 (32x) \\
   &     &   & 1+8+28+49+35+7 (64x) \\
   &     &   & 1+8+28+52+35+4 (16x) \\
88 & 128 & 6 & 1+8+28+48+34+8+1 (56x) & 10752 & 8+56+64 & not selfdual\\ % #0098
   &     &   & 1+8+28+49+35+7 (64x) \\
   &     &   & 1+8+28+56+35 (8x) \\
89 & 132 & 6 & 1+8+28+47+36+11+1 (8x) & 16 & 9*4+8*8+2*16 & \\ % #0075
   &     &   & 1+8+28+48+36+10+1 (8x) \\
   &     &   & 1+8+28+47+37+11 (12x) \\
   &     &   & 1+8+28+48+37+10 (20x) \\
   &     &   & 1+8+28+49+37+9 (36x) \\
   &     &   & 1+8+28+50+37+8 (48x) \\
90 & 132 & 6 & 1+8+28+48+36+10+1 (36x) & 72 & 2*12+36+72 & \\ % #0088
   &     &   & 1+8+28+50+36+8+1 (12x) \\
   &     &   & 1+8+28+49+37+9 (72x) \\
   &     &   & 1+8+28+50+37+8 (12x) \\
91 & 144 & 6 & 1+8+28+48+41+16+2 & 384 & 48+96 & $2^2\times\Delta_{6.6}$ \\ % #0078
92 & 144 & 6 & 1+8+28+48+41+16+2 & 768 & 48+96 & $2^2\times\Delta_{6.4}$ \\ % #0080
93 & 144 & 6 & 1+8+28+48+41+16+2 & 34560 & tra & $2^2\times\Delta_{6.5}$ \\ % #0084
94 & 144 & 6 & 1+8+28+49+42+15+1 & 24192 & tra & $2\times\Delta_{7.29}$ \\ % #0100
95 & 160 & 6 & 1+8+28+50+47+22+4 & 384 & 32+32+96 & $2^2\times\Delta_{6.7}$ \\ % #0076
96 & 160 & 6 & 1+8+28+50+47+22+4 & 960 & tra & $2^2\times\Delta_{6.8}$ \\ % #0079
97 & 160 & 6 & 1+8+28+51+48+21+3 (64x) & 96 & 16+48+96 & $2\times\Delta_{7.33}$ \\ % #0087
   &     &   & 1+8+28+52+49+20+2 (96x) \\
98 & 164 & 6 & 1+8+28+50+48+24+5 (20x) & 512 & 4+2*16+2*32+64 & \\ % #0091
   &     &   & 1+8+28+51+49+23+4 (16x) \\
   &     &   & 1+8+28+52+50+22+3 (32x) \\
   &     &   & 1+8+28+53+52+21+1 (32x) \\
   &     &   & 1+8+28+53+53+21 (64x) \\
99 & 176 & 6 & 1+8+28+52+53+28+6 & 63360 & tra & $2^3\times\Delta_{5.1}$ \\ % #0085
100 & 192 & 7 & 1+8+28+53+57+34+10+1 & 23040 & tra & $2^3\times\Delta_{5.2}$ \\ % #0086
101 & 192 & 7 & 1+8+28+53+57+34+10+1 (64x) & 2048 & 64+128 & $2^2\times\Delta_{6.10}$ \\ % #0092
   &     &   & 1+8+28+54+59+34+8 (128x) \\
102 & 196 & 6 & 1+8+28+54+60+36+9 & 225792 & tra & $\Delta_{4.1}\times\Delta_{4.1}$\\ % #0090
103 & 224 & 7 & 1+8+28+55+65+46+18+3 & 129024 & tra & $2^4\times\Delta_{4.1}$ \\ % #0089
104 & 256 & 8 & 1+8+28+56+70+56+28+8+1 & 10321920 & tra & $2^8$ \\ % #0102
\end{supertabular}}

\subsection{Universal covers}
Of the graphs given here only $\Delta_{8.34}$, $\Delta_{8.38}$,
$\Delta_{8.71}$, $\Delta_{8.74}$, $\Delta_{8.75}$ are not their own
universal $(0,2)$-cover.
(The universal $(0,2)$-cover of the first two is $2^3 \times \Delta_{5.2}$,
that of the last three is $2^8$.)

\subsection{Spectra}
There are two pairs and one triple of cospectral graphs here,
namely $\Delta_{8.37}$, $\Delta_{8.38}$ and $\Delta_{8.40}$, $\Delta_{8.41}$
and $\Delta_{8.72}$, $\Delta_{8.73}$, $\Delta_{8.74}$.

Seven graphs have integral spectrum, namely $\Delta_{8.13}$
(spectrum $\pm 8^1$ $\pm 4^{10}$ $\pm 2^{24}$ $0^{10}$),
$\Delta_{8.71}$ (spectrum $\pm 8^1$ $\pm 4^{28}$ $0^{70}$),
$\Delta_{8.72}$, $\Delta_{8.73}$, $\Delta_{8.74}$
(spectrum $\pm 8^1$ $\pm 6^2$ $\pm 4^{16}$ $\pm 2^{30}$ $0^{30}$),
$\Delta_{8.75}$
(spectrum $\pm 8^1$ $\pm 6^1$ $\pm 4^{22}$ $\pm 2^{15}$ $0^{50}$),
and $\Delta_{8.104}$
(spectrum $\pm 8^1$ $\pm 6^8$ $\pm 4^{28}$ $\pm 2^{56}$ $0^{70}$).

Among the bipartite $(0,2)$-graphs of valency 8,
one has $2\sqrt{2} \le \lambda_2 \le 6$, where the smallest value
is attained only for $\Delta_{8.2}$, and quite a few graphs have $\lambda_2 = 6$.

\medskip

\section{Nonbipartite $(0,2)$-graphs with $k = 8$}
The nonbipartite $(0,2)$-graphs with $k = 8$ have bipartite doubles
found above. The number $N(k)$ of nonisomorphic non-bipartite $(0,2)$-graphs
with $k \le 8$ is given in the following table.

\medskip\begin{center}
\begin{tabular}{|c|rrrrrrrrr|}
\hline
$k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
$N(k)$ & 0 & 0 & 0 & 1 & 1 & 4 & 11 & 56 & 198 \\
\hline
\end{tabular}
\end{center}

The nonbipartite $(0,2)$-graph of valency $k$ with serial number $i$
is called $\Gamma_{k.i}$.

\medskip
In one case we give spectral information in order to distinguish two graphs
with otherwise identical data; $\lambda_i$ denotes the $i$-th largest
eigenvalue.
We sometimes note in the tables that graphs are cospectral
(abbreviated `cosp'); such information is given only when there was
no other remark in the `graph' column. See also \S\ref{spec}.
Many of these graphs can be found by adding a matching to a $(0,2)$-graph
of smaller valency. See \S\ref{match}.

\medskip{\footnotesize
\tablehead{
\# & bipd & $k$ & $v$ & $d$ & distribution & locally & gsz & orbits & graph \\
\hline}
\tabletail{\hline}
\noindent\begin{supertabular}{|r|r|rrrllrll}
1 & 8.3 & 8 & 34 & 3 & 1+8+21+4 (10x) & C71 & 2 & 17*2 & \\ % n034.d3.nr003.q0001
 & & & & & 1+8+22+3 (14x) & C611 & \\ 
 & & & & & 1+8+23+2 (10x) & C5111 & \\ 
2 & 8.3 & 8 & 34 & 3 & 1+8+20+5 (2x) & C8 & 2 & 17*2 & \\ % n034.d3.nr003.q0002
 & & & & & 1+8+21+4 (8x) & C71 & \\ 
 & & & & & 1+8+22+3 (12x) & C611 & \\ 
 & & & & & 1+8+23+2 (12x) & C5111 & \\ 
3 & 8.4 & 8 & 34 & 3 & 1+8+20+5 (2x) & C53 & 10 & 2*2+3*10 & \\ % n034.d3.nr004.q0001
 & & & & & 1+8+22+3 (20x) & C611 & \\ 
 & & & & & 1+8+23+2 (10x) & C5111 & \\ 
 & & & & & 1+8+25 (2x) & C311111 & \\ 
4 & 8.7 & 8 & 36 & 3 & 1+8+22+5 & C3311 & 72 & tra & \\ % s8.103.q0007
5 & 8.7 & 8 & 36 & 3 & 1+8+22+5 (12x) & C611 & 12 & 12+12+12 & \\ % s8.103.q0002
 & & & & & 1+8+23+4 (24x) & C5111 & \\ 
6 & 8.7 & 8 & 36 & 3 & 1+8+20+7 (12x) & C8 & 12 & 4*6+12 & \\ % s8.103.q0001
 & & & & & 1+8+22+5 (12x) & C611 & \\ 
 & & & & & 1+8+23+4 (12x) & C5111 & \\
7 & 8.8 & 8 & 37 & 3 & 1+8+20+8 (6x) & C8 & 12 & 2*1+2+3+3*6+12 & \\ % n037.d3.nr007.q0001
 & & & & & 1+8+22+6 (16x) & C611 & \\ 
 & & & & & 1+8+23+5 (12x) & C5111 & \\ 
 & & & & & 1+8+28 (3x) & C11111111 & \\ 
8 & 8.8 & 8 & 37 & 3 & 1+8+22+6 (28x) & C3311 & 336 & 1+8+28 & \\ % n037.d3.nr007.q0023
 & & & & & 1+8+28 (9x) & C11111111 & \\ 
9 & 8.13 & 8 & 40 & 3 & 1+8+28+3 & recta & 1920 & tra & cf. \S\ref{aq}, \ref{gq}\\ % s8.104.q0063
10 & 8.13 & 8 & 40 & 3 & 1+8+22+9 (32x) & C3311 & 384 & 8+32 & cosp $\Gamma_{8.16}$ \\ % s8.104.q0014
 & & & & & 1+8+28+3 (8x) & C11111111 & \\ 
11 & 8.11 & 8 & 40 & 3 & 1+8+22+9 (20x) & C3311 & 10 & 4*10 & \\ % n040.d3.nr010.q0001
 & & & & & 1+8+25+6 (20x) & C311111 & \\ 
12 & 8.12 & 8 & 40 & 3 & 1+8+20+11 (16x) & C8 & 32 & 8+16+16 & \\ % n040.d3.nr011.q0001
 & & & & & 1+8+21+10 (16x) & C71 & \\ 
 & & & & & 1+8+22+9 (8x) & C611 & \\ 
13 & 8.12 & 8 & 40 & 3 & 1+8+22+9 (24x) & C3311 & 96 & 16+24 & \\ % n040.d3.nr011.q0005
 & & & & & 1+8+25+6 (16x) & C311111 & \\ 
14 & 8.15 & 8 & 40 & 3 & 1+8+25+6 (32x) & C311111 & 192 & 8+32 & \\ % n040.d3.nr013.q0005
 & & & & & 1+8+28+3 (8x) & C11111111 & \\ 
15 & 8.13 & 8 & 40 & 4 & 1+8+20+10+1 (24x) & C8 & 192 & 16+24 & \\ % s8.104.q0002
 & & & & & 1+8+22+9 (16x) & C611 & \\
16 & 8.13 & 8 & 40 & 4 & 1+8+20+10+1 (8x) & C8 & 32 & 3*8+16 & cosp $\Gamma_{8.10}$ \\ % s8.104.q0001
 & & & & & 1+8+22+9 (8x) & C611 & \\ 
 & & & & & 1+8+23+8 (16x) & C5111 & \\ 
 & & & & & 1+8+28+3 (8x) & C11111111 & \\ 
17 & 8.15 & 8 & 40 & 4 & 1+8+20+10+1 (8x) & C8 & 64 & 8+32 & \\ % n040.d4.nr013.q0001
 & & & & & 1+8+21+10 (32x) & C71 & \\ 
18 & 8.24 & 8 & 42 & 3 & 1+8+25+8 (32x) & C311111 & 8 & 2+6*4+2*8 & \\ % n042.d3.nr022.q0002
 & & & & & 1+8+28+5 (10x) & C11111111 & \\ 
19 & 8.25 & 8 & 42 & 3 & 1+8+21+12 (16x) & C71 & 8 & 3*2+3*4+3*8 & \\ % n042.d3.nr023.q0001
 & & & & & 1+8+22+11 (4x) & C611 & \\ 
 & & & & & 1+8+23+10 (16x) & C5111 & \\ 
 & & & & & 1+8+28+5 (6x) & C11111111 & \\ 
20 & 8.25 & 8 & 42 & 3 & 1+8+22+11 (4x) & C3311 & 8 & 3*2+3*4+3*8 & \\ % n042.d3.nr023.q0004
 & & & & & 1+8+25+8 (24x) & C311111 & \\ 
 & & & & & 1+8+28+5 (14x) & C11111111 & \\ 
21 & 8.29 & 8 & 42 & 3 & 1+8+21+12 (24x) & C71 & 48 & 6+12+24 & cosp $\Gamma_{8.31}$ \\ % n042.d3.nr027.q0005
 & & & & & 1+8+22+11 (12x) & C611 & \\ 
 & & & & & 1+8+28+5 (6x) & C11111111 & \\ 
22 & 8.29 & 8 & 42 & 3 & 1+8+22+11 (4x) & C3311 & 16 & 2+2*4+2*8+16 & \\ % n042.d3.nr027.q0006
 & & & & & 1+8+25+8 (24x) & C311111 & \\ 
 & & & & & 1+8+28+5 (14x) & C11111111 & \\ 
23 & 8.24 & 8 & 42 & 4 & 1+8+20+12+1 (2x) & C8 & 8 & 2+6*4+2*8 & \\ % n042.d4.nr022.q0001
 & & & & & 1+8+21+11+1 (8x) & C71 & \\ 
 & & & & & 1+8+21+12 (12x) & C71 & \\ 
 & & & & & 1+8+22+11 (8x) & C611 & \\ 
 & & & & & 1+8+23+10 (12x) & C5111 & \\ 
24 & 8.25 & 8 & 42 & 4 & 1+8+20+12+1 (2x) & C8 & 8 & 3*2+3*4+3*8 & \\ % n042.d4.nr023.q0002
 & & & & & 1+8+22+10+1 (12x) & C611 & \\ 
 & & & & & 1+8+21+12 (4x) & C71 & \\ 
 & & & & & 1+8+22+11 (4x) & C611 & \\ 
 & & & & & 1+8+23+10 (20x) & C5111 & \\ 
25 & 8.25 & 8 & 42 & 4 & 1+8+20+12+1 (2x) & C8 & 8 & 3*2+3*4+3*8 & \\ % n042.d4.nr023.q0003
 & & & & & 1+8+22+10+1 (4x) & C611 & \\ 
 & & & & & 1+8+21+12 (8x) & C71 & \\ 
 & & & & & 1+8+22+11 (2x) & C611 & \\ 
 & & & & & 1+8+22+11 (2x) & C3311 & \\ 
 & & & & & 1+8+23+10 (12x) & C5111 & \\ 
 & & & & & 1+8+25+8 (12x) & C311111 & \\ 
26 & 8.26 & 8 & 42 & 4 & 1+8+20+12+1 (2x) & C8 & 8 & 3*2+3*4+3*8 & cosp $\Gamma_{8.28}$ \\ % n042.d4.nr024.q0001
 & & & & & 1+8+20+12+1 (4x) & C53 & \\ 
 & & & & & 1+8+22+10+1 (4x) & C611 & \\ 
 & & & & & 1+8+21+12 (8x) & C71 & \\ 
 & & & & & 1+8+22+11 (2x) & C611 & \\ 
 & & & & & 1+8+22+11 (2x) & C3311 & \\ 
 & & & & & 1+8+23+10 (8x) & C5111 & \\ 
 & & & & & 1+8+25+8 (8x) & C311111 & \\ 
 & & & & & 1+8+28+5 (4x) & C11111111 & \\ 
27 & 8.26 & 8 & 42 & 4 & 1+8+20+12+1 (2x) & C8 & 32 & 2+8+2*16 & \\ % n042.d4.nr024.q0002
 & & & & & 1+8+22+10+1 (16x) & C611 & \\ 
 & & & & & 1+8+22+11 (8x) & C611 & \\ 
 & & & & & 1+8+23+10 (16x) & C5111 & \\ 
28 & 8.26 & 8 & 42 & 4 & 1+8+20+12+1 (4x) & C8 & 8 & 3*2+3*4+3*8 & cosp $\Gamma_{8.26}$ \\ % n042.d4.nr024.q0004
 & & & & & 1+8+22+10+1 (4x) & C611 & \\ 
 & & & & & 1+8+21+12 (8x) & C71 & \\ 
 & & & & & 1+8+22+11 (4x) & C611 & \\ 
 & & & & & 1+8+23+10 (16x) & C5111 & \\ 
 & & & & & 1+8+28+5 (6x) & C11111111 & \\ 
29 & 8.26 & 8 & 42 & 4 & 1+8+22+10+1 (8x) & C3311 & 32 & 2+8+2*16 & \\ % n042.d4.nr024.q0009
 & & & & & 1+8+25+8 (16x) & C311111 & \\ 
 & & & & & 1+8+28+5 (18x) & C11111111 & \\ 
30 & 8.29 & 8 & 42 & 4 & 1+8+20+12+1 (14x) & C8 & 16 & 2+2*4+2*8+16 & \\ % n042.d4.nr027.q0001
 & & & & & 1+8+22+10+1 (4x) & C611 & \\ 
 & & & & & 1+8+21+12 (16x) & C71 & \\ 
 & & & & & 1+8+23+10 (8x) & C5111 & \\ 
31 & 8.29 & 8 & 42 & 4 & 1+8+20+12+1 (6x) & C8 & 48 & 6+12+24 & cosp $\Gamma_{8.21}$ \\ % n042.d4.nr027.q0002
 & & & & & 1+8+22+10+1 (12x) & C611 & \\ 
 & & & & & 1+8+23+10 (24x) & C5111 & \\ 
32 & 8.31 & 8 & 44 & 3 & 1+8+22+13 (6x) & C611 & 4 & 6*2+8*4 & \\ % n044.d3.nr029.q0001
 & & & & & 1+8+22+13 (2x) & C3311 & \\ 
 & & & & & 1+8+23+12 (12x) & C5111 & \\ 
 & & & & & 1+8+25+10 (20x) & C311111 & \\ 
 & & & & & 1+8+28+7 (4x) & C11111111 & \\ 
33 & 8.32 & 8 & 44 & 3 & 1+8+22+13 (6x) & C611 & 4 & 6*2+8*4 & \\ % n044.d3.nr030.q0001
 & & & & & 1+8+22+13 (2x) & C3311 & \\ 
 & & & & & 1+8+23+12 (12x) & C5111 & \\ 
 & & & & & 1+8+25+10 (20x) & C311111 & \\ 
 & & & & & 1+8+28+7 (4x) & C11111111 & \\ 
34 & 8.33 & 8 & 48 & 3 & 1+8+23+16 (24x) & C5111 & 48 & 24+24 & \\ % n048.d3.nr031.q0001
 & & & & & 1+8+25+14 (24x) & C311111 & \\ 
35 & 8.34 & 8 & 48 & 3 & 1+8+23+16 & C5111 & 2880 & tra & cf. \S\ref{aq} \\ % n048.d3.nr032.q0027
36 & 8.34 & 8 & 48 & 3 & 1+8+23+16 (24x) & C5111 & 480 & 24+24 & \\ % n048.d3.nr032.q0028
 & & & & & 1+8+28+11 (24x) & C11111111 & \\ 
37 & 8.34 & 8 & 48 & 3 & 1+8+28+11 & recta & 960 & tra & \\ % n048.d3.nr032.q0041
38 & 8.37 & 8 & 48 & 3 & 1+8+23+16 & C5111 & 192 & tra & $2\times\Gamma_{7.1}$ \\ % n048.d3.nr035.q0003
39 & 8.37 & 8 & 48 & 3 & 1+8+23+16 & C5111 & 192 & tra & $2\times\Gamma_{7.2}$ \\ % n048.d3.nr035.q0004
40 & 8.37 & 8 & 48 & 3 & 1+8+25+14 (32x) & C311111 & 32 & 16+32 & cosp $\Gamma_{8.46}$ \\ % n048.d3.nr035.q0008
 & & & & & 1+8+28+11 (16x) & C11111111 & \\ 
41 & 8.37 & 8 & 48 & 3 & 1+8+25+14 (16x) & C311111 & 64 & 16+32 & \\ % n048.d3.nr035.q0014
 & & & & & 1+8+28+11 (32x) & C11111111 & \\ 
42 & 8.37 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 192 & tra & cosp $\Gamma_{8.45}$ \\ % n048.d3.nr035.q0015
43 & 8.38 & 8 & 48 & 3 & 1+8+23+16 & C5111 & 960 & tra & $2\times\Gamma_{7.4}$ \\ % n048.d3.nr036.q0007
44 & 8.38 & 8 & 48 & 3 & 1+8+23+16 & C5111 & 960 & tra & $2\times\Gamma_{7.3}$ \\ % n048.d3.nr036.q0008
45 & 8.38 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 96 & tra & cosp $\Gamma_{8.42}$ \\ % n048.d3.nr036.q0030
46 & 8.38 & 8 & 48 & 3 & 1+8+25+14 (32x) & C311111 & 64 & 16+32 & cosp $\Gamma_{8.40}$ \\ % n048.d3.nr036.q0031
 & & & & & 1+8+28+11 (16x) & C11111111 & \\ 
47 & 8.38 & 8 & 48 & 3 & 1+8+28+11 & recta & 960 & tra & \\ % n048.d3.nr036.q0086
48 & 8.39 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 32 & 16+32 & \\ % n048.d3.nr037.q0005
49 & 8.39 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 96 & tra & \\ % n048.d3.nr037.q0006
50 & 8.40 & 8 & 48 & 3 & 1+8+25+14 (32x) & C311111 & 16 & 2*4+8+2*16 & cosp $\Gamma_{8.53}$ \\ % n048.d3.nr038.q0003
 & & & & & 1+8+28+11 (16x) & C11111111 & \\ 
51 & 8.40 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 32 & 16+32 & \\ % n048.d3.nr038.q0005
52 & 8.41 & 8 & 48 & 3 & 1+8+25+14 (16x) & C311111 & 16 & 2*4+8+2*16 & \\ % n048.d3.nr039.q0003
 & & & & & 1+8+28+11 (32x) & C11111111 & \\ 
53 & 8.41 & 8 & 48 & 3 & 1+8+25+14 (32x) & C311111 & 32 & 16+32 & cosp $\Gamma_{8.50}$ \\ % n048.d3.nr039.q0004
 & & & & & 1+8+28+11 (16x) & C11111111 & \\ 
54 & 8.42 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 48 & tra & $\lambda_2 = 4$, $\lambda_3 = 3.909$ \\ % n048.d3.nr040.q0007
55 & 8.42 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 48 & tra & $\lambda_2 = 4$, $\lambda_3 = 3.646$ \\ % n048.d3.nr040.q0008
56 & 8.42 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 48 & tra & cosp $\Gamma_{8.54}$ \\ % n048.d3.nr040.q0009
57 & 8.43 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 24 & 12+12+24 & \\ % n048.d3.nr041.q0003
58 & 8.44 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 96 & tra & \\ % n048.d3.nr042.q0001
59 & 8.45 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 48 & tra & \\ % n048.d3.nr043.q0001
60 & 8.50 & 8 & 48 & 3 & 1+8+25+14 (24x) & C311111 & 48 & 24+24 & \\ % n048.d3.nr048.q0003
 & & & & & 1+8+28+11 (24x) & C11111111 & \\ 
61 & 8.52 & 8 & 48 & 3 & 1+8+25+14 & C311111 & 96 & tra & \\ % n048.d3.nr050.q0002
62 & 8.34 & 8 & 48 & 4 & 1+8+20+16+3 & C53 & 2880 & tra & $K_4\times\Gamma_{5.1}$ \\ % n048.d4.nr032.q0001
63 & 8.34 & 8 & 48 & 4 & 1+8+22+15+2 (16x) & C3311 & 64 & 16+16+16 & \\ % n048.d4.nr032.q0002
 & & & & & 1+8+25+14 (16x) & C311111 & \\ 
 & & & & & 1+8+28+11 (16x) & C11111111 & \\ 
64 & 8.36 & 8 & 48 & 4 & 1+8+20+18+1 (2x) & C53 & 4 & 6*2+9*4 & \\ % n048.d4.nr034.q0001
 & & & & & 1+8+21+17+1 (2x) & C71 & \\ 
 & & & & & 1+8+22+16+1 (2x) & C3311 & \\ 
 & & & & & 1+8+25+13+1 (2x) & C311111 & \\ 
 & & & & & 1+8+23+16 (12x) & C5111 & \\ 
 & & & & & 1+8+25+14 (24x) & C311111 & \\ 
 & & & & & 1+8+28+11 (4x) & C11111111 & \\ 
65 & 8.36 & 8 & 48 & 4 & 1+8+21+17+1 (6x) & C71 & 12 & 2*6+3*12 & \\ % n048.d4.nr034.q0004
 & & & & & 1+8+22+16+1 (12x) & C611 & \\ 
 & & & & & 1+8+23+15+1 (6x) & C5111 & \\ 
 & & & & & 1+8+23+16 (24x) & C5111 & \\ 
66 & 8.37 & 8 & 48 & 4 & 1+8+25+12+2 & C311111 & 384 & tra & cosp $\Gamma_{8.69}$ \\ % n048.d4.nr035.q0001
67 & 8.37 & 8 & 48 & 4 & 1+8+25+12+2 (16x) & C311111 & 128 & 16+32 & cosp $\Gamma_{8.70}$ \\ % n048.d4.nr035.q0002
 & & & & & 1+8+25+14 (32x) & C311111 & \\ 
68 & 8.37 & 8 & 48 & 4 & 1+8+22+16+1 & C3311 & 192 & tra & $2\times\Gamma_{7.5}$ \\ % n048.d4.nr035.q0010
69 & 8.38 & 8 & 48 & 4 & 1+8+25+12+2 & C311111 & 1920 & tra & cosp $\Gamma_{8.66}$ \\ % n048.d4.nr036.q0001
70 & 8.38 & 8 & 48 & 4 & 1+8+25+12+2 (16x) & C311111 & 128 & 16+32 & cosp $\Gamma_{8.67}$ \\ % n048.d4.nr036.q0002
 & & & & & 1+8+25+14 (32x) & C311111 & \\ 
71 & 8.38 & 8 & 48 & 4 & 1+8+22+16+1 & C611 & 96 & tra & $2\times\Gamma_{7.6}$ \\ % n048.d4.nr036.q0010
72 & 8.39 & 8 & 48 & 4 & 1+8+22+16+1 (24x) & C611 & 24 & 4*12 & $2\times\Gamma_{7.11}$ \\ % n048.d4.nr037.q0001
 & & & & & 1+8+23+16 (24x) & C5111 & \\ 
73 & 8.40 & 8 & 48 & 4 & 1+8+21+16+2 (8x) & C71 & 16 & 4*8+16 & $2\times\Gamma_{7.7}$ \\ % n048.d4.nr038.q0001
 & & & & & 1+8+22+16+1 (32x) & C611 & \\ 
 & & & & & 1+8+23+16 (8x) & C5111 & \\ 
74 & 8.41 & 8 & 48 & 4 & 1+8+21+16+2 (16x) & C71 & 32 & 16+16+16 & $2\times\Gamma_{7.8}$ \\ % n048.d4.nr039.q0001
 & & & & & 1+8+22+16+1 (16x) & C611 & \\ 
 & & & & & 1+8+23+16 (16x) & C5111 & \\ 
75 & 8.42 & 8 & 48 & 4 & 1+8+21+16+2 (8x) & C71 & 8 & 6*8 & $2\times\Gamma_{7.9}$ \\ % n048.d4.nr040.q0001
 & & & & & 1+8+22+16+1 (32x) & C611 & \\ 
 & & & & & 1+8+23+16 (8x) & C5111 & \\ 
76 & 8.43 & 8 & 48 & 4 & 1+8+21+16+2 (24x) & C71 & 24 & 4*12 & $2\times\Gamma_{7.10}$ \\ % n048.d4.nr041.q0001
 & & & & & 1+8+22+16+1 (24x) & C611 & \\ 
77 & 8.50 & 8 & 48 & 4 & 1+8+21+17+1 (8x) & C71 & 16 & 4*8+16 & \\ % n048.d4.nr048.q0001
 & & & & & 1+8+22+16+1 (8x) & C611 & \\ 
 & & & & & 1+8+23+15+1 (24x) & C5111 & \\ 
 & & & & & 1+8+23+16 (8x) & C5111 & \\ 
78 & 8.50 & 8 & 48 & 4 & 1+8+22+16+1 (24x) & C611 & 24 & 12+12+24 & \\ % n048.d4.nr048.q0004
 & & & & & 1+8+23+16 (24x) & C5111 & \\ 
79 & 8.51 & 8 & 48 & 4 & 1+8+21+17+1 (6x) & C71 & 12 & 2*6+3*12 & \\ % n048.d4.nr049.q0001
 & & & & & 1+8+23+15+1 (42x) & C5111 & \\ 
80 & 8.51 & 8 & 48 & 4 & 1+8+21+17+1 (6x) & C71 & 12 & 6*6+12 & \\ % n048.d4.nr049.q0002
 & & & & & 1+8+22+16+1 (12x) & C611 & \\ 
 & & & & & 1+8+23+15+1 (6x) & C5111 & \\ 
 & & & & & 1+8+23+16 (24x) & C5111 & \\ 
81 & 8.52 & 8 & 48 & 4 & 1+8+21+16+2 & C71 & 672 & tra & $2\times\Gamma_{7.12}$ \\ % n048.d4.nr050.q0001
82 & 8.58 & 8 & 52 & 4 & 1+8+25+17+1 (8x) & C311111 & 8 & 3*4+5*8 & \\ % n052.d4.nr056.q0001
 & & & & & 1+8+25+18 (32x) & C311111 & \\ 
 & & & & & 1+8+28+15 (12x) & C11111111 & \\ 
83 & 8.58 & 8 & 52 & 4 & 1+8+22+20+1 (4x) & C3311 & 16 & 3*4+3*8+16 & \\ % n052.d4.nr056.q0003
 & & & & & 1+8+25+18 (32x) & C311111 & \\ 
 & & & & & 1+8+28+15 (16x) & C11111111 & \\ 
84 & 8.58 & 8 & 52 & 4 & 1+8+22+19+2 (8x) & C611 & 16 & 3*4+3*8+16 & \\ % n052.d4.nr056.q0004
 & & & & & 1+8+23+18+2 (16x) & C5111 & \\ 
 & & & & & 1+8+22+20+1 (12x) & C611 & \\ 
 & & & & & 1+8+23+20 (8x) & C5111 & \\ 
 & & & & & 1+8+25+18 (8x) & C311111 & \\ 
85 & 8.59 & 8 & 52 & 4 & 1+8+25+15+3 (2x) & C311111 & 12 & 2*2+8*6 & \\ % n052.d4.nr057.q0001
 & & & & & 1+8+22+20+1 (12x) & C611 & \\ 
 & & & & & 1+8+23+19+1 (12x) & C5111 & \\ 
 & & & & & 1+8+28+14+1 (6x) & C11111111 & \\ 
 & & & & & 1+8+25+18 (6x) & C311111 & \\ 
 & & & & & 1+8+28+15 (14x) & C11111111 & \\ 
86 & 8.59 & 8 & 52 & 4 & 1+8+22+20+1 (4x) & C3311 & 8 & 5*4+4*8 & \\ % n052.d4.nr057.q0002
 & & & & & 1+8+28+14+1 (4x) & C11111111 & \\ 
 & & & & & 1+8+25+18 (32x) & C311111 & \\ 
 & & & & & 1+8+28+15 (12x) & C11111111 & \\ 
87 & 8.60 & 8 & 52 & 4 & 1+8+23+18+2 (24x) & C5111 & 48 & 2+6+8+12+24 & \\ % n052.d4.nr058.q0001
 & & & & & 1+8+22+20+1 (12x) & C611 & \\ 
 & & & & & 1+8+22+21 (8x) & C611 & \\ 
 & & & & & 1+8+25+18 (8x) & C311111 & \\ 
88 & 8.60 & 8 & 52 & 4 & 1+8+22+20+1 (4x) & C3311 & 16 & 2*2+2*4+3*8+16 & \\ % n052.d4.nr058.q0002
 & & & & & 1+8+25+18 (32x) & C311111 & \\ 
 & & & & & 1+8+28+15 (16x) & C11111111 & \\ 
89 & 8.61 & 8 & 52 & 4 & 1+8+20+21+2 (6x) & C8 & 48 & 2+3*6+8+24 & \\ % n052.d4.nr059.q0001
 & & & & & 1+8+23+18+2 (24x) & C5111 & \\ 
 & & & & & 1+8+28+14+1 (6x) & C11111111 & \\ 
 & & & & & 1+8+25+18 (8x) & C311111 & \\ 
 & & & & & 1+8+28+15 (8x) & C11111111 & \\ 
90 & 8.63 & 8 & 56 & 4 & 1+8+25+18+4 (4x) & C311111 & 8 & 4*4+5*8 & \\ % n056.d4.nr061.q0001
 & & & & & 1+8+25+20+2 (32x) & C311111 & \\ 
 & & & & & 1+8+28+17+2 (8x) & C11111111 & \\ 
 & & & & & 1+8+25+22 (4x) & C311111 & \\ 
 & & & & & 1+8+28+19 (8x) & C11111111 & \\ 
91 & 8.63 & 8 & 56 & 4 & 1+8+25+18+4 (4x) & C311111 & 8 & 4*4+5*8 & \\ % n056.d4.nr061.q0002
 & & & & & 1+8+25+20+2 (8x) & C311111 & \\ 
 & & & & & 1+8+25+22 (12x) & C311111 & \\ 
 & & & & & 1+8+28+19 (32x) & C11111111 & \\ 
92 & 8.66 & 8 & 56 & 4 & 1+8+25+20+2 (16x) & C311111 & 16 & 3*8+2*16 & \\ % n056.d4.nr064.q0001
 & & & & & 1+8+25+22 (8x) & C311111 & \\ 
 & & & & & 1+8+28+19 (32x) & C11111111 & \\ 
93 & 8.66 & 8 & 56 & 4 & 1+8+25+20+2 (16x) & C311111 & 8 & 7*8 & \\ % n056.d4.nr064.q0002
 & & & & & 1+8+25+22 (8x) & C311111 & \\ 
 & & & & & 1+8+28+19 (32x) & C11111111 & \\ 
94 & 8.67 & 8 & 56 & 4 & 1+8+25+18+4 (4x) & C311111 & 16 & 2*4+2*8+2*16 & \\ % n056.d4.nr065.q0001
 & & & & & 1+8+25+20+2 (32x) & C311111 & \\ 
 & & & & & 1+8+28+17+2 (16x) & C11111111 & \\ 
 & & & & & 1+8+25+22 (4x) & C311111 & \\ 
95 & 8.68 & 8 & 56 & 4 & 1+8+25+18+4 (4x) & C311111 & 16 & 2*4+2*8+2*16 & \\ % n056.d4.nr066.q0001
 & & & & & 1+8+25+20+2 (16x) & C311111 & \\ 
 & & & & & 1+8+25+22 (4x) & C311111 & \\ 
 & & & & & 1+8+28+19 (32x) & C11111111 & \\ 
96 & 8.69 & 8 & 56 & 4 & 1+8+25+20+2 (16x) & C311111 & 4 & 14*4 & \\ % n056.d4.nr067.q0001
 & & & & & 1+8+28+17+2 (8x) & C11111111 & \\ 
 & & & & & 1+8+25+22 (8x) & C311111 & \\ 
 & & & & & 1+8+28+19 (24x) & C11111111 & \\ 
97 & 8.71 & 8 & 64 & 3 & 1+8+25+30 & C311111 & 46080 & tra & $K_4\times\Gamma_{5.2}$ \\ % n064.d3.nr069.q0001
98 & 8.71 & 8 & 64 & 3 & 1+8+25+30 & C311111 & 4608 & tra & cf. \S\ref{fq} \\ % n064.d3.nr069.q0002
99 & 8.72 & 8 & 64 & 3 & 1+8+25+30 (48x) & C311111 & 256 & 16+16+32 & \\ % n064.d3.nr070.q0019
 & & & & & 1+8+28+27 (16x) & C11111111 & \\ 
100 & 8.72 & 8 & 64 & 3 & 1+8+25+30 (16x) & C311111 & 128 & 16+16+32 & cosp $\Gamma_{8.101}$ \\ % n064.d3.nr070.q0021
 & & & & & 1+8+28+27 (48x) & C11111111 & \\ 
101 & 8.72 & 8 & 64 & 3 & 1+8+25+30 (16x) & C311111 & 256 & 16+16+32 & cosp $\Gamma_{8.100}$ \\ % n064.d3.nr070.q0022
 & & & & & 1+8+28+27 (48x) & C11111111 & \\ 
102 & 8.73 & 8 & 64 & 3 & 1+8+25+30 (32x) & C311111 & 128 & 32+32 & cosp $\Gamma_{8.103}$ \\ % n064.d3.nr071.q0005
 & & & & & 1+8+28+27 (32x) & C11111111 & \\ 
103 & 8.73 & 8 & 64 & 3 & 1+8+25+30 (32x) & C311111 & 64 & 4*16 & cosp $\Gamma_{8.102}$ \\ % n064.d3.nr071.q0013
 & & & & & 1+8+28+27 (32x) & C11111111 & \\ 
104 & 8.74 & 8 & 64 & 3 & 1+8+25+30 (32x) & C311111 & 512 & 32+32 & cosp $\Gamma_{8.102}$ \\ % n064.d3.nr072.q0211
 & & & & & 1+8+28+27 (32x) & C11111111 & \\ 
105 & 8.74 & 8 & 64 & 3 & 1+8+25+30 & C311111 & 15360 & tra & $2^8/\langle 1^30^5,01^6\rangle$\\ % n064.d3.nr072.q0213
106 & 8.74 & 8 & 64 & 3 & 1+8+25+30 (32x) & C311111 & 768 & 32+32 & cosp $\Gamma_{8.102}$ \\ % n064.d3.nr072.q0215
 & & & & & 1+8+28+27 (32x) & C11111111 & \\ 
107 & 8.74 & 8 & 64 & 3 & 1+8+28+27 & recta & 1536 & tra & cosp $\Gamma_{8.108}$ \\ % n064.d3.nr072.q0519
108 & 8.74 & 8 & 64 & 3 & 1+8+28+27 & recta & 9216 & tra & $2^8/\langle 1^50^3,0^21^6\rangle$\\ % n064.d3.nr072.q0523
109 & 8.75 & 8 & 64 & 3 & 1+8+25+30 & C311111 & 7680 & 32+32 & \\ % n064.d3.nr073.q0021
110 & 8.75 & 8 & 64 & 3 & 1+8+25+30 (48x) & C311111 & 384 & 4*16 & \\ % n064.d3.nr073.q0022
 & & & & & 1+8+28+27 (16x) & C11111111 & \\ 
111 & 8.75 & 8 & 64 & 3 & 1+8+25+30 (32x) & C311111 & 2304 & 32+32 & \\ % n064.d3.nr073.q0087
 & & & & & 1+8+28+27 (32x) & C11111111 & \\ 
112 & 8.77 & 8 & 64 & 3 & 1+8+28+27 & recta & 1152 & tra & \\ % n064.d3.nr075.q0022
113 & 8.72 & 8 & 64 & 4 & 1+8+25+24+6 & C311111 & 1024 & tra & $2\times\Gamma_{7.18}$ \\ % n064.d4.nr070.q0001
114 & 8.72 & 8 & 64 & 4 & 1+8+25+24+6 & C311111 & 512 & tra & $2\times\Gamma_{7.19}$ \\ % n064.d4.nr070.q0002
115 & 8.73 & 8 & 64 & 4 & 1+8+25+24+6 & C311111 & 1536 & tra & $2\times\Gamma_{7.21}$ \\ % n064.d4.nr071.q0001
116 & 8.73 & 8 & 64 & 4 & 1+8+25+24+6 & C311111 & 192 & tra & $2\times\Gamma_{7.20}$ \\ % n064.d4.nr071.q0003
117 & 8.74 & 8 & 64 & 4 & 1+8+22+24+9 & C611 & 1536 & tra & $2^2\times\Gamma_{6.2}$ \\ % n064.d4.nr072.q0001
118 & 8.74 & 8 & 64 & 4 & 1+8+22+24+9 & C3311 & 9216 & tra & $2^2\times K_4 \times K_4$ \\ % n064.d4.nr072.q0002
119 & 8.74 & 8 & 64 & 4 & 1+8+22+24+9 (32x) & C611 & 768 & 32+32 & cosp $\Gamma_{8.113-116,119-122}$ \\ % n064.d4.nr072.q0008
 & & & & & 1+8+28+27 (32x) & C11111111 & \\ 
120 & 8.74 & 8 & 64 & 4 & 1+8+22+24+9 (32x) & C3311 & 4608 & 32+32 & cosp $\Gamma_{8.113-116,119-122}$ \\ % n064.d4.nr072.q0009
 & & & & & 1+8+28+27 (32x) & C11111111 & \\ 
121 & 8.74 & 8 & 64 & 4 & 1+8+25+24+6 & C311111 & 3072 & tra & $2\times\Gamma_{7.25}$ \\ % n064.d4.nr072.q0057
122 & 8.74 & 8 & 64 & 4 & 1+8+25+24+6 & C311111 & 512 & tra & $2\times\Gamma_{7.26}$ \\ % n064.d4.nr072.q0058
123 & 8.75 & 8 & 64 & 4 & 1+8+22+24+9 (32x) & C3311 & 2304 & 32+32 & $K_4 \times \Gamma_{5.4}$ \\ % n064.d4.nr073.q0001
 & & & & & 1+8+25+30 (32x) & C311111 & \\ 
124 & 8.76 & 8 & 64 & 4 & 1+8+22+24+9 (16x) & C611 & 16 & 4*16 & $2\times\Gamma_{7.16}$ \\ % n064.d4.nr074.q0001
 & & & & & 1+8+23+24+8 (48x) & C5111 & \\ 
125 & 8.76 & 8 & 64 & 4 & 1+8+25+26+4 (16x) & C311111 & 16 & 4*16 & \\ % n064.d4.nr074.q0002
 & & & & & 1+8+25+28+2 (16x) & C311111 & \\ 
 & & & & & 1+8+28+25+2 (32x) & C11111111 & \\ 
126 & 8.76 & 8 & 64 & 4 & 1+8+25+24+6 & C311111 & 48 & 16+48 & $2\times\Gamma_{7.17}$ \\ % n064.d4.nr074.q0004
127 & 8.76 & 8 & 64 & 4 & 1+8+25+26+4 (32x) & C311111 & 16 & 4*16 & \\ % n064.d4.nr074.q0005
 & & & & & 1+8+28+25+2 (32x) & C11111111 & \\ 
128 & 8.76 & 8 & 64 & 4 & 1+8+28+25+2 (48x) & recta & 48 & 16+48 & \\ % n064.d4.nr074.q0010
 & & & & & 1+8+28+27 (16x) &  & \\ 
129 & 8.77 & 8 & 64 & 4 & 1+8+22+24+9 (16x) & C611 & 96 & 16+48 & $2\times\Gamma_{7.24}$ \\ % n064.d4.nr075.q0001
 & & & & & 1+8+23+24+8 (48x) & C5111 & \\ 
130 & 8.77 & 8 & 64 & 4 & 1+8+25+26+4 (32x) & C311111 & 64 & 32+32 & \\ % n064.d4.nr075.q0007
 & & & & & 1+8+28+25+2 (32x) & C11111111 & \\ 
131 & 8.77 & 8 & 64 & 4 & 1+8+28+25+2 (48x) & recta & 96 & 16+48 & \\ % n064.d4.nr075.q0023
 & & & & & 1+8+28+27 (16x) &  & \\ 
132 & 8.78 & 8 & 64 & 4 & 1+8+25+26+4 (16x) & C311111 & 8 & 8*8 & \\ % n064.d4.nr076.q0001
 & & & & & 1+8+25+28+2 (8x) & C311111 & \\ 
 & & & & & 1+8+28+25+2 (24x) & C11111111 & \\ 
 & & & & & 1+8+25+30 (8x) & C311111 & \\ 
 & & & & & 1+8+28+27 (8x) & C11111111 & \\ 
133 & 8.79 & 8 & 64 & 4 & 1+8+21+24+10 (4x) & C71 & 8 & 8*4+4*8 & $2\times\Gamma_{7.14}$ \\ % n064.d4.nr077.q0001
 & & & & & 1+8+22+24+9 (4x) & C611 & \\ 
 & & & & & 1+8+22+24+9 (4x) & C3311 & \\ 
 & & & & & 1+8+23+24+8 (28x) & C5111 & \\ 
 & & & & & 1+8+25+24+6 (16x) & C311111 & \\ 
 & & & & & 1+8+28+24+3 (8x) & C11111111 & \\ 
134 & 8.79 & 8 & 64 & 4 & 1+8+25+24+6 (4x) & C311111 & 8 & 8*4+4*8 & \\ % n064.d4.nr077.q0002
 & & & & & 1+8+25+25+5 (4x) & C311111 & \\ 
 & & & & & 1+8+25+26+4 (16x) & C311111 & \\ 
 & & & & & 1+8+25+27+3 (8x) & C311111 & \\ 
 & & & & & 1+8+28+24+3 (4x) & C11111111 & \\ 
 & & & & & 1+8+28+25+2 (16x) & C11111111 & \\ 
 & & & & & 1+8+28+27 (12x) & C11111111 & \\ 
135 & 8.80 & 8 & 64 & 4 & 1+8+25+23+7 (4x) & C311111 & 8 & 8*4+4*8 & \\ % n064.d4.nr078.q0002
 & & & & & 1+8+25+25+5 (8x) & C311111 & \\ 
 & & & & & 1+8+25+26+4 (4x) & C311111 & \\ 
 & & & & & 1+8+25+27+3 (8x) & C311111 & \\ 
 & & & & & 1+8+25+28+2 (8x) & C311111 & \\ 
 & & & & & 1+8+28+25+2 (12x) & C11111111 & \\ 
 & & & & & 1+8+28+27 (20x) & C11111111 & \\ 
136 & 8.81 & 8 & 64 & 4 & 1+8+25+26+4 (24x) & C311111 & 4 & 16*4 & \\ % n064.d4.nr079.q0001
 & & & & & 1+8+25+28+2 (8x) & C311111 & \\ 
 & & & & & 1+8+28+25+2 (24x) & C11111111 & \\ 
 & & & & & 1+8+28+27 (8x) & C11111111 & \\ 
137 & 8.81 & 8 & 64 & 4 & 1+8+22+24+9 (12x) & C611 & 12 & 4+5*12 & $2\times\Gamma_{7.13}$ \\ % n064.d4.nr079.q0003
 & & & & & 1+8+23+24+8 (24x) & C5111 & \\ 
 & & & & & 1+8+25+24+6 (24x) & C311111 & \\ 
 & & & & & 1+8+28+24+3 (4x) & C11111111 & \\ 
138 & 8.82 & 8 & 64 & 4 & 1+8+21+24+10 (2x) & C71 & 4 & 8*2+12*4 & $2\times\Gamma_{7.15}$ \\ % n064.d4.nr080.q0001
 & & & & & 1+8+22+24+9 (8x) & C611 & \\ 
 & & & & & 1+8+22+24+9 (2x) & C3311 & \\ 
 & & & & & 1+8+23+24+8 (26x) & C5111 & \\ 
 & & & & & 1+8+25+24+6 (20x) & C311111 & \\ 
 & & & & & 1+8+28+24+3 (6x) & C11111111 & \\ 
139 & 8.83 & 8 & 64 & 4 & 1+8+25+26+4 (16x) & C311111 & 16 & 2*8+3*16 & \\ % n064.d4.nr081.q0004
 & & & & & 1+8+28+24+3 (8x) & C11111111 & \\ 
 & & & & & 1+8+25+28+2 (16x) & C311111 & \\ 
 & & & & & 1+8+28+25+2 (24x) & C11111111 & \\ 
140 & 8.83 & 8 & 64 & 4 & 1+8+28+24+3 (8x) & recta & 112 & 8+56 & \\ % n064.d4.nr081.q0005
 & & & & & 1+8+28+25+2 (56x) &  & \\ 
141 & 8.83 & 8 & 64 & 4 & 1+8+22+24+9 (56x) & C611 & 336 & 8+56 & $2\times\Gamma_{7.23}$ \\ % n064.d4.nr081.q0028
 & & & & & 1+8+28+24+3 (8x) & C11111111 & \\ 
142 & 8.83 & 8 & 64 & 4 & 1+8+22+24+9 (8x) & C3311 & 48 & 8+8+48 & $2\times\Gamma_{7.22}$ \\ % n064.d4.nr081.q0029
 & & & & & 1+8+25+24+6 (48x) & C311111 & \\ 
 & & & & & 1+8+28+24+3 (8x) & C11111111 & \\ 
143 & 8.80 & 8 & 64 & 5 & 1+8+21+23+10+1 (12x) & C71 & 24 & 4+5*12 & $2\times\Gamma_{7.29}$ \\ % n064.d5.nr078.q0001
 & & & & & 1+8+23+23+8+1 (12x) & C5111 & \\ 
 & & & & & 1+8+23+24+8 (24x) & C5111 & \\ 
 & & & & & 1+8+25+24+6 (16x) & C311111 & \\ 
144 & 8.83 & 8 & 64 & 5 & 1+8+21+23+10+1 (8x) & C71 & 112 & 8+56 & $2\times\Gamma_{7.30}$ \\ % n064.d5.nr081.q0001
 & & & & & 1+8+23+24+8 (56x) & C5111 & \\ 
145 & 8.91 & 8 & 72 & 4 & 1+8+25+28+10 (40x) & C311111 & 16 & 3*8+3*16 & $2\times\Gamma_{7.32}$ \\ % n072.d4.nr089.q0001
 & & & & & 1+8+28+28+7 (32x) & C11111111 & \\ 
146 & 8.91 & 8 & 72 & 4 & 1+8+25+32+6 (24x) & C311111 & 32 & 2*4+4*8+2*16 & cosp $\Gamma_{8.147}$ \\ % n072.d4.nr089.q0005
 & & & & & 1+8+28+29+6 (8x) & C11111111 & \\ 
 & & & & & 1+8+28+32+3 (32x) & C11111111 & \\ 
 & & & & & 1+8+28+35 (8x) & C11111111 & \\ 
147 & 8.91 & 8 & 72 & 4 & 1+8+25+32+6 (24x) & C311111 & 32 & 3*8+3*16 & cosp $\Gamma_{8.146}$ \\ % n072.d4.nr089.q0006
 & & & & & 1+8+28+32+3 (48x) & C11111111 & \\ 
148 & 8.91 & 8 & 72 & 4 & 1+8+25+32+6 (24x) & C311111 & 192 & 24+48 & \\ % n072.d4.nr089.q0014
 & & & & & 1+8+28+29+6 (48x) & C11111111 & \\ 
149 & 8.91 & 8 & 72 & 4 & 1+8+25+32+6 (24x) & C311111 & 192 & 24+48 & \\ % n072.d4.nr089.q0020
 & & & & & 1+8+28+35 (48x) & C11111111 & \\ 
150 & 8.92 & 8 & 72 & 4 & 1+8+25+28+10 (40x) & C311111 & 64 & 8+2*16+32 & $2\times\Gamma_{7.31}$ \\ % n072.d4.nr090.q0001
 & & & & & 1+8+28+28+7 (32x) & C11111111 & \\ 
151 & 8.92 & 8 & 72 & 4 & 1+8+25+32+6 (24x) & C311111 & 32 & 5*8+2*16 & cosp $\Gamma_{8.152}$ \\ % n072.d4.nr090.q0003
 & & & & & 1+8+28+29+6 (8x) & C11111111 & \\ 
 & & & & & 1+8+28+32+3 (32x) & C11111111 & \\ 
 & & & & & 1+8+28+35 (8x) & C11111111 & \\ 
152 & 8.92 & 8 & 72 & 4 & 1+8+25+32+6 (24x) & C311111 & 96 & 24+24+24 & cosp $\Gamma_{8.151}$ \\ % n072.d4.nr090.q0008
 & & & & & 1+8+28+29+6 (24x) & C11111111 & \\ 
 & & & & & 1+8+28+35 (24x) & C11111111 & \\ 
153 & 8.93 & 8 & 72 & 4 & 1+8+25+32+6 (24x) & C311111 & 160 & 4+8+20+40 & \\ % n072.d4.nr091.q0001
 & & & & & 1+8+28+32+3 (48x) & C11111111 & \\ 
154 & 8.95 & 8 & 80 & 4 & 1+8+25+34+12 (32x) & C311111 & 32 & 6*8+2*16 & cosp $\Gamma_{8.156}$ \\ % n080.d4.nr093.q0009
 & & & & & 1+8+28+31+12 (8x) & C11111111 & \\ 
 & & & & & 1+8+28+34+9 (8x) & C11111111 & \\ 
 & & & & & 1+8+28+37+6 (8x) & C11111111 & \\ 
 & & & & & 1+8+28+40+3 (24x) & C11111111 & \\ 
155 & 8.95 & 8 & 80 & 4 & 1+8+28+31+12 (16x) & recta & 192 & 16+16+48 & \\ % n080.d4.nr093.q0012
 & & & & & 1+8+28+34+9 (48x) &  & \\ 
 & & & & & 1+8+28+40+3 (16x) &  & \\ 
156 & 8.95 & 8 & 80 & 4 & 1+8+25+34+12 (32x) & C311111 & 64 & 3*16+32 & cosp $\Gamma_{8.154}$ \\ % n080.d4.nr093.q0014
 & & & & & 1+8+28+37+6 (48x) & C11111111 & \\ 
157 & 8.96 & 8 & 80 & 4 & 1+8+25+34+12 (32x) & C311111 & 32 & 5*16 & \\ % n080.d4.nr094.q0009
 & & & & & 1+8+28+31+12 (16x) & C11111111 & \\ 
 & & & & & 1+8+28+34+9 (16x) & C11111111 & \\ 
 & & & & & 1+8+28+37+6 (16x) & C11111111 & \\ 
158 & 8.96 & 8 & 80 & 4 & 1+8+28+40+3 & recta & 480 & tra & \\ % n080.d4.nr094.q0049
159 & 8.97 & 8 & 80 & 4 & 1+8+25+33+13 (16x) & C311111 & 16 & 4*8+3*16 & \\ % n080.d4.nr095.q0001
 & & & & & 1+8+28+32+11 (8x) & C11111111 & \\ 
 & & & & & 1+8+25+36+10 (16x) & C311111 & \\ 
 & & & & & 1+8+28+36+7 (24x) & C11111111 & \\ 
 & & & & & 1+8+28+39+4 (16x) & C11111111 & \\ 
160 & 8.97 & 8 & 80 & 4 & 1+8+25+33+13 (16x) & C311111 & 16 & 4*8+3*16 & \\ % n080.d4.nr095.q0002
 & & & & & 1+8+25+34+12 (16x) & C311111 & \\ 
 & & & & & 1+8+28+32+11 (8x) & C11111111 & \\ 
 & & & & & 1+8+28+33+10 (16x) & C11111111 & \\ 
 & & & & & 1+8+28+36+7 (24x) & C11111111 & \\ 
161 & 8.95 & 8 & 80 & 5 & 1+8+22+28+17+4 (16x) & C611 & 192 & 16+16+48 & $2^2\times\Gamma_{6.4}$ \\ % n080.d5.nr093.q0001
 & & & & & 1+8+23+29+16+3 (48x) & C5111 & \\ 
 & & & & & 1+8+25+31+14+1 (16x) & C311111 & \\ 
162 & 8.95 & 8 & 80 & 5 & 1+8+22+28+17+4 (8x) & C611 & 96 & 4*8+2*24 & \\ % n080.d5.nr093.q0002
 & & & & & 1+8+23+29+16+3 (24x) & C5111 & \\ 
 & & & & & 1+8+25+31+14+1 (8x) & C311111 & \\ 
 & & & & & 1+8+28+31+12 (8x) & C11111111 & \\ 
 & & & & & 1+8+28+34+9 (24x) & C11111111 & \\ 
 & & & & & 1+8+28+40+3 (8x) & C11111111 & \\ 
163 & 8.95 & 8 & 80 & 5 & 1+8+25+30+14+2 (16x) & C311111 & 32 & 3*16+32 & $2\times\Gamma_{7.34}$ \\ % n080.d5.nr093.q0003
 & & & & & 1+8+25+32+14 (32x) & C311111 & \\ 
 & & & & & 1+8+28+32+11 (32x) & C11111111 & \\ 
164 & 8.96 & 8 & 80 & 5 & 1+8+25+30+14+2 (16x) & C311111 & 16 & 5*16 & $2\times\Gamma_{7.35}$ \\ % n080.d5.nr094.q0001
 & & & & & 1+8+25+32+14 (32x) & C311111 & \\ 
 & & & & & 1+8+28+32+11 (32x) & C11111111 & \\ 
165 & 8.96 & 8 & 80 & 5 & 1+8+25+31+14+1 & C311111 & 480 & tra & $2^2\times\Gamma_{6.3}$ \\ % n080.d5.nr094.q0010
166 & 8.96 & 8 & 80 & 5 & 1+8+25+31+14+1 (40x) & C311111 & 240 & 40+40 & \\ % n080.d5.nr094.q0011
 & & & & & 1+8+28+40+3 (40x) & C11111111 & \\ 
167 & 8.99 & 8 & 88 & 4 & 1+8+25+36+18 (40x) & C311111 & 160 & 8+40+40 & $2\times\Gamma_{7.37}$ \\ % n088.d4.nr097.q0001
 & & & & & 1+8+28+36+15 (48x) & C11111111 & \\ 
168 & 8.99 & 8 & 88 & 4 & 1+8+25+36+18 & C311111 & 31680 & tra & $K_4\times\Delta_{5.1}$ \\ % n088.d4.nr097.q0091
169 & 8.99 & 8 & 88 & 4 & 1+8+25+36+18 (24x) & C311111 & 576 & 16+24+48 & \\ % n088.d4.nr097.q0092
 & & & & & 1+8+28+42+9 (64x) & C11111111 & \\ 
170 & 8.100 & 8 & 96 & 4 & 1+8+28+43+16 (64x) & recta & 768 & 32+64 & \\ % n096.d4.nr098.q0256
 & & & & & 1+8+28+53+6 (32x) &  & \\ 
171 & 8.100 & 8 & 96 & 4 & 1+8+28+53+6 & recta & 11520 & tra & cf. \S\ref{aq} \\ % n096.d4.nr098.q0345
172 & 8.100 & 8 & 96 & 5 & 1+8+25+35+22+5 (64x) & C311111 & 256 & 32+64 & $2^2\times\Gamma_{6.5}$ \\ % n096.d5.nr098.q0005
 & & & & & 1+8+28+38+19+2 (32x) & C11111111 & \\ 
173 & 8.100 & 8 & 96 & 5 & 1+8+25+35+22+5 (32x) & C311111 & 128 & 2*16+2*32 & \\ % n096.d5.nr098.q0006
 & & & & & 1+8+28+38+19+2 (16x) & C11111111 & \\ 
 & & & & & 1+8+28+43+16 (32x) & C11111111 & \\ 
 & & & & & 1+8+28+53+6 (16x) & C11111111 & \\ 
174 & 8.100 & 8 & 96 & 5 & 1+8+25+37+22+3 (48x) & C311111 & 192 & 48+48 & $2\times\Gamma_{7.42}$ \\ % n096.d5.nr098.q0033
 & & & & & 1+8+28+40+19 (48x) & C11111111 & \\ 
175 & 8.100 & 8 & 96 & 5 & 1+8+25+37+22+3 (32x) & C311111 & 128 & 32+32+32 & $2\times\Gamma_{7.43}$ \\ % n096.d5.nr098.q0035
 & & & & & 1+8+28+37+19+3 (32x) & C11111111 & \\ 
 & & & & & 1+8+28+40+19 (32x) & C11111111 & \\ 
176 & 8.100 & 8 & 96 & 5 & 1+8+25+37+22+3 & C311111 & 11520 & tra & $K_4\times\Delta_{5.2}$ \\ % n096.d5.nr098.q0129
177 & 8.100 & 8 & 96 & 5 & 1+8+25+37+22+3 (32x) & C311111 & 768 & 32+64 & \\ % n096.d5.nr098.q0130
 & & & & & 1+8+28+43+16 (64x) & C11111111 & \\ 
178 & 8.100 & 8 & 96 & 5 & 1+8+28+38+19+2 & recta & 3840 & tra & $2^2\times\Gamma_{6.6}$ \\ % n096.d5.nr098.q0141
179 & 8.100 & 8 & 96 & 5 & 1+8+28+38+19+2 (48x) & recta & 1920 & 48+48 & \\ % n096.d5.nr098.q0145
 & & & & & 1+8+28+53+6 (48x) &  & \\ 
180 & 8.100 & 8 & 96 & 5 & 1+8+28+37+19+3 & recta & 1920 & tra & $2\times\Gamma_{7.44}$ \\ % n096.d5.nr098.q0233
181 & 8.101 & 8 & 96 & 5 & 1+8+25+37+22+3 (16x) & C311111 & 128 & 2*8+16+2*32 & \\ % n096.d5.nr099.q0001
 & & & & & 1+8+25+38+24 (32x) & C311111 & \\ 
 & & & & & 1+8+28+40+19 (16x) & C11111111 & \\ 
 & & & & & 1+8+28+41+18 (32x) & C11111111 & \\ 
182 & 8.101 & 8 & 96 & 5 & 1+8+25+37+22+3 (16x) & C311111 & 64 & 4*8+2*16+32 & \\ % n096.d5.nr099.q0002
 & & & & & 1+8+25+38+24 (16x) & C311111 & \\ 
 & & & & & 1+8+28+40+19 (8x) & C11111111 & \\ 
 & & & & & 1+8+28+41+18 (16x) & C11111111 & \\ 
 & & & & & 1+8+28+44+15 (32x) & C11111111 & \\ 
 & & & & & 1+8+28+53+6 (8x) & C11111111 & \\ 
183 & 8.101 & 8 & 96 & 5 & 1+8+25+38+22+2 (32x) & C311111 & 64 & 32+32+32 & $2\times\Gamma_{7.38}$ \\ % n096.d5.nr099.q0003
 & & & & & 1+8+28+38+19+2 (32x) & C11111111 & \\ 
 & & & & & 1+8+28+40+19 (32x) & C11111111 & \\ 
184 & 8.101 & 8 & 96 & 5 & 1+8+25+35+22+5 (16x) & C311111 & 64 & 4*16+32 & $2\times\Gamma_{7.39}$ \\ % n096.d5.nr099.q0013
 & & & & & 1+8+25+36+22+4 (16x) & C311111 & \\ 
 & & & & & 1+8+28+37+19+3 (16x) & C11111111 & \\ 
 & & & & & 1+8+28+38+19+2 (48x) & C11111111 & \\ 
185 & 8.101 & 8 & 96 & 5 & 1+8+25+35+22+5 (32x) & C311111 & 256 & 32+64 & $2\times\Gamma_{7.40}$ \\ % n096.d5.nr099.q0045
 & & & & & 1+8+28+38+19+2 (64x) & C11111111 & \\ 
186 & 8.100 & 8 & 96 & 6 & 1+8+23+32+23+8+1 & C5111 & 5760 & tra & $2^3\times\Gamma_{5.1}$ \\ % n096.d6.nr098.q0001
187 & 8.100 & 8 & 96 & 6 & 1+8+23+32+23+8+1 (48x) & C5111 & 960 & 48+48 & $2\times\Gamma_{7.47}$ \\ % n096.d6.nr098.q0002
 & & & & & 1+8+28+37+19+3 (48x) & C11111111 & \\ 
188 & 8.103 & 8 & 112 & 4 & 1+8+28+45+30 (64x) & recta & 2304 & 16+48+48 & \\ % n112.d4.nr101.q0407
 & & & & & 1+8+28+55+20 (48x) &  & \\ 
189 & 8.103 & 8 & 112 & 5 & 1+8+25+39+30+9 (64x) & C311111 & 384 & 16+48+48 & $2^2\times\Gamma_{6.8}$ \\ % n112.d5.nr101.q0001
 & & & & & 1+8+28+42+27+6 (48x) & C11111111 & \\ 
190 & 8.103 & 8 & 112 & 5 & 1+8+25+39+30+9 (32x) & C311111 & 192 & 2*8+4*24 & \\ % n112.d5.nr101.q0003
 & & & & & 1+8+28+42+27+6 (24x) & C11111111 & \\ 
 & & & & & 1+8+28+45+30 (32x) & C11111111 & \\ 
 & & & & & 1+8+28+55+20 (24x) & C11111111 & \\ 
191 & 8.103 & 8 & 112 & 5 & 1+8+25+39+30+9 & C311111 & 16128 & tra & $2\times K_4\times\Delta_{4.1}$ \\ % n112.d5.nr101.q0193
192 & 8.103 & 8 & 112 & 5 & 1+8+25+39+30+9 (48x) & C311111 & 768 & 16+32+64 & $2\times\Gamma_{7.50}$ \\ % n112.d5.nr101.q0194
 & & & & & 1+8+28+45+27+3 (64x) & C11111111 & \\ 
193 & 8.104 & 8 & 128 & 4 & 1+8+28+56+35 & recta & 645120 & tra & $2\times\Gamma_{7.51}$ \\ % n128.d4.nr102.q1961
194 & 8.104 & 8 & 128 & 5 & 1+8+28+46+35+10 & recta & 92160 & tra & $2^3\times\Gamma_{5.2}$ \\ % n128.d5.nr102.q0286
195 & 8.104 & 8 & 128 & 5 & 1+8+28+46+35+10 (64x) & recta & 15360 & 64+64 & $2\times\Gamma_{7.53}$ \\ % n128.d5.nr102.q0298
 & & & & & 1+8+28+56+35 (64x) &  & \\ 
196 & 8.104 & 8 & 128 & 6 & 1+8+25+40+35+16+3 & C311111 & 92160 & tra & $2^5\times K_4$ \\ % n128.d6.nr102.q0001
197 & 8.104 & 8 & 128 & 6 & 1+8+25+40+35+16+3 (64x) & C311111 & 4608 & 64+64 & $2^3\times\Gamma_{5.4}$ \\ % n128.d6.nr102.q0002
 & & & & & 1+8+28+46+35+10 (64x) & C11111111 & \\ 
198 & 8.104 & 8 & 128 & 6 & 1+8+25+40+35+16+3 (32x) & C311111 & 1536 & 32+32+64 & $2\times\Gamma_{7.56}$ \\ % n128.d6.nr102.q0006
 & & & & & 1+8+28+46+35+10 (64x) & C11111111 & \\ 
 & & & & & 1+8+28+56+35 (32x) & C11111111 & \\ 

\end{supertabular}}

\subsection{Recognition}
All graphs are uniquely determined by the combinatorial data given above, with the exception
of the three graphs $\Gamma_{8.54}$, $\Gamma_{8.55}$, $\Gamma_{8.56}$.
The spectrum distinguishes $\Gamma_{8.55}$ from the other two, but
$\Gamma_{8.54}$ and $\Gamma_{8.56}$ are also cospectral. They can be
distinguished as follows:
Fix a vertex $\infty$. Vertices at distance two can be identified with
pairs of vertices in $\Omega := \Gamma(\infty)$, that is, with edges
in the complete graph on $\Omega$. Vertices at distance three from $\infty$
then give rise to  a graph on $\Omega$ that is regular of valency 2.
We can list the frequencies with which the various possibilities
($C_8$, $C_7$, $C_6$, $C_4$, $C_3$, $2C_4$, $C_4+C_3$, $2C_3$) occur.
Each of these three graphs has a transitive group, so the
choice of $\infty$ does not matter.
For $\Gamma_{8.54}$: 3*$C_7$, 3*$C_6$, 2*$C_5$, 2*$C_3$, 1*$(C_4+C_3)$, 3*$(2C_3)$.
For $\Gamma_{8.55}$: 2*$C_7$, 1*$C_6$, 2*$C_5$, 2*$C_3$, 2*$(C_4+C_3)$, 5*$(2C_3)$.
For $\Gamma_{8.56}$: 1*$C_7$, 2*$C_6$, 1*$C_5$, 2*$C_3$, 2*$(C_4+C_3)$, 6*$(2C_3)$.

\subsection{Spectra}\label{spec}
Among the nonbipartite $(0,2)$-graphs of valency 8, one has
$2 \le \lambda_2 \le 6$, where the smallest value is attained
only for $\Gamma_{8.9}$, and quite a few graphs have $\lambda_2 = 6$.

Graphs with integral spectrum (listed are the multiplicities for the
eigenvalues 8, 6, ..., $-6$):

\medskip{\footnotesize
\begin{tabular}{|r|c|rrrrrrrr|}
\hline
$v$ & graph & 8 & 6 & 4 & 2 & 0 & $-2$ & $-4$ & $-6$ \\
\hline
40 & $\Gamma_{8.9}$                   & 1 & 0 &  0 & 20 & 5 & 4 & 10 & 0 \\
40 & $\Gamma_{8.i}$, $i = 10, 16$     & 1 & 0 &  4 & 12 & 5 & 12 & 6 & 0 \\
40 & $\Gamma_{8.15}$                  & 1 & 0 &  6 &  8 & 5 & 16 & 4 & 0 \\
\hline
64 & $\Gamma_{8.i}$, $i = 97, 98$     & 1 & 0 & 13 &  0 & 35 & 0 & 15 & 0 \\
64 & $\Gamma_{8.99}$                  & 1 & 0 & 10 & 12 & 15 & 18 & 6 & 2 \\
64 & $\Gamma_{8.i}$, $i = 100, 101$   & 1 & 0 &  8 & 16 & 15 & 14 & 8 & 2 \\
64 & $\Gamma_{8.i}$, $i = 102, 103, 104, 106$
                                      & 1 & 0 &  9 & 14 & 15 & 16 & 7 & 2 \\
64 & $\Gamma_{8.105}$                 & 1 & 0 & 11 & 10 & 15 & 20 & 5 & 2 \\
64 & $\Gamma_{8.i}$, $i = 107, 108$   & 1 & 0 &  7 & 18 & 15 & 12 & 9 & 2 \\
64 & $\Gamma_{8.109}$                 & 1 & 0 & 12 &  5 & 25 & 10 & 10 & 1 \\
64 & $\Gamma_{8.110}$                 & 1 & 0 & 11 &  7 & 25 &  8 & 11 & 1 \\
64 & $\Gamma_{8.111}$                 & 1 & 0 & 10 &  9 & 25 &  6 & 12 & 1 \\
64 & $\Gamma_{8.i}$, $i=113,114,115,116,119,120,121,122$
                                      & 1 & 1 &  7 & 15 & 15 & 15 &  9 & 1 \\
64 & $\Gamma_{8.i}$, $i=117,118$      & 1 & 2 &  7 & 12 & 15 & 18 &  9 & 0 \\
64 & $\Gamma_{8.123}$                 & 1 & 1 & 10 &  6 & 25 &  9 & 12 & 0 \\
\hline
128 & $\Gamma_{8.193}$                 & 1 & 1 & 21 & 21 & 35 & 35 &  7 & 7 \\
128 & $\Gamma_{8.194}$                 & 1 & 3 & 13 & 31 & 35 & 25 & 15 & 5 \\
128 & $\Gamma_{8.195}$                 & 1 & 2 & 17 & 26 & 35 & 30 & 11 & 6 \\
128 & $\Gamma_{8.196}$                 & 1 & 5 & 13 & 25 & 35 & 31 & 15 & 3 \\
128 & $\Gamma_{8.197}$                 & 1 & 4 & 13 & 28 & 35 & 28 & 15 & 4 \\
128 & $\Gamma_{8.198}$                 & 1 & 3 & 15 & 27 & 35 & 29 & 13 & 5 \\
\hline
\end{tabular}}

\medskip
(Note that all integral eigenvalues of a $(0,2)$-graph have the same
parity as the valency $k$, since the adjacency matrix $A$ satisfies
$A^2 = kI + 2B$ for some integral matrix $B$.)

\medskip
Apart from the cases already noted in the table, and those apparent from
this eigenvalue table, there are two cospectral pairs and one cospectral quadruple:
$\Gamma_{8.68}$, $\Gamma_{8.71}$ and $\Gamma_{8.73}$, $\Gamma_{8.74}$ and
$\Gamma_{8.38}$, $\Gamma_{8.39}$, $\Gamma_{8.43}$, $\Gamma_{8.44}$.

\medskip
\section{Some constructions}

\subsection{Adding a matching}\label{match}
Let $\Gamma$ be a $(0,2)$-graph of valency $k$, and let $\sigma$
be an automorphism of order 2 such that $x$ and $\sigma(x)$ have distance
different from 0, 1 or 3 for all vertices $x$. Add edges
$(i,\sigma(i))$ to $\Gamma$ to obtain a graph $\Delta$
of valency $k+1$ on the same number of vertices.
If $\Gamma$ is bipartite, or, more generally, if whenever $a$ and $b$
are two distinct vertices with two common neighbours, $\sigma(a)$
is nonadjacent to $b$, then $\Delta$ is a $(0,2)$-graph again.

On can go in the opposite direction: given a $(0,2)$-graph of valency $k+1$
and an automorphism $\sigma$ of which all orbits are edges. Removing these
edges yields a graph of which all connected components are $(0,2)$-graphs
of valency $k$.

In the situation discussed here, the eigenvalues of $\Delta$
are those of $\Gamma$ shifted by $\pm 1$. Indeed, for each eigenvalue
$\theta$ of $\Gamma$, the $\theta$-eigenspace has a basis of vectors $u$
with $\sigma(u) = \pm u$. (For arbitrary $u$, consider $u \pm \sigma(u)$.)
Such a $u$ is also eigenvector of $\Delta$, with eigenvalue $\theta \pm 1$.

\medskip
This construction allows one to go back and forth between
$\Delta_{2.1}$ (the quadrangle) and $\Gamma_{3.1}$ ($K_4$),
and for example between $\Delta_{7.1}$ and $\Gamma_{8.i}$,
$i=40,41,42,66,67$. We give the correspondences in the table below.

\noindent
\begin{tabular}{|cc|cc|cc|}
\hline

$\Delta_{3.1}$ & $\Gamma_{4.1}$ &
$\Delta_{7.8}$ & $\Gamma_{8.57}$ &
$\Delta_{7.36}$ & $\Gamma_{8.167-169}$ \\

$\Delta_{4.2}$ & $\Gamma_{5.2-4}$ &
$\Delta_{7.9}$ & $\Gamma_{8.48-49}$ &
$\Delta_{7.37}$ & $\Gamma_{8.181-185}$ \\

$\Delta_{5.2}$ & $\Gamma_{6.5-6}$ &
$\Delta_{7.10}$ & $\Gamma_{8.61}$ &
$\Delta_{7.38}$ & $\Delta_{8.34,38}$, $\Gamma_{8.170-180}$ \\

$\Delta_{5.3}$ & $\Gamma_{6.8}$ &
$\Delta_{7.11}$ & $\Gamma_{8.94}$ &
$\Delta_{7.39}$ & $\Gamma_{8.188-192}$ \\

$\Delta_{5.4}$ & $\Delta_{6.3}$, $\Gamma_{6.9-11}$ &
$\Delta_{7.12}$ & $\Gamma_{8.90-91}$ &
$\Delta_{7.40}$ & $\Delta_{8.71,74-75}$, $\Gamma_{8.193-198}$ \\

$\Delta_{6.1}$ & $\Gamma_{7.18-19}$ &
$\Delta_{7.13}$ & $\Gamma_{8.95}$ &
$\Gamma_{5.3}$ & $\Gamma_{6.1}$ \\

$\Delta_{6.2}$ & $\Gamma_{7.20-21}$ &
$\Delta_{7.16}$ & $\Gamma_{8.96}$ &
$\Gamma_{6.7}$ & $\Gamma_{7.3}$ \\

$\Delta_{6.3}$ & $\Gamma_{7.25-26}$ &
$\Delta_{7.17}$ & $\Gamma_{8.92-93}$ &
$\Gamma_{6.9}$ & $\Gamma_{7.25}$ \\

$\Delta_{6.4}$ & $\Gamma_{7.31}$ &
$\Delta_{7.18}$ & $\Gamma_{8.125,127-128}$ &
$\Gamma_{6.10}$ & $\Gamma_{7.25,27}$ \\

$\Delta_{6.6}$ & $\Gamma_{7.32}$ &
$\Delta_{7.19}$ & $\Gamma_{8.132}$ &
$\Gamma_{6.11}$ & $\Gamma_{7.26}$ \\

$\Delta_{6.7}$ & $\Gamma_{7.34}$ &
$\Delta_{7.20}$ & $\Gamma_{8.99-101,113-114}$ &
$\Gamma_{7.41}$ & $\Gamma_{8.63,70}$ \\

$\Delta_{6.8}$ & $\Gamma_{7.35}$ &
$\Delta_{7.21}$ & $\Gamma_{8.102-103,115-116}$ &
$\Gamma_{7.42}$ & $\Gamma_{8.45}$ \\

$\Delta_{6.9}$ & $\Gamma_{7.37}$ &
$\Delta_{7.22}$ & $\Gamma_{8.112,130-131}$ &
$\Gamma_{7.43}$ & $\Gamma_{8.46,63}$ \\

$\Delta_{6.10}$ & $\Gamma_{7.38-40}$ &
$\Delta_{7.23}$ & $\Gamma_{8.104-108,121-122}$ &
$\Gamma_{7.44}$ & $\Gamma_{8.37,47}$ \\

$\Delta_{6.11}$ & $\Delta_{7.2}$, $\Gamma_{7.41-45}$ &
$\Delta_{7.24}$ & $\Gamma_{8.136}$ &
$\Gamma_{7.45}$ & $\Gamma_{8.37,69}$ \\

$\Delta_{6.12}$ & $\Gamma_{7.48-50}$ &
$\Delta_{7.25}$ & $\Gamma_{8.135}$ &
$\Gamma_{7.46}$ & $\Gamma_{8.35,44,62}$ \\

$\Delta_{6.13}$ & $\Delta_{7.23}$, $\Gamma_{7.51-56}$ &
$\Delta_{7.26}$ & $\Gamma_{8.134}$ &
$\Gamma_{7.47}$ & $\Gamma_{8.36}$ \\

$\Delta_{7.1}$ & $\Gamma_{8.40-42,66-67}$ &
$\Delta_{7.28}$ & $\Gamma_{8.139-140}$ &
$\Gamma_{7.51}$ & $\Gamma_{8.105}$ \\

$\Delta_{7.2}$ & $\Gamma_{8.45-47,69-70}$ &
$\Delta_{7.30}$ & $\Gamma_{8.150-152}$ &
$\Gamma_{7.52}$ & $\Gamma_{8.97,108,111,121}$ \\

$\Delta_{7.3}$ & $\Gamma_{8.59}$ &
$\Delta_{7.31}$ & $\Gamma_{8.153}$ &
$\Gamma_{7.53}$ & $\Gamma_{8.106,109}$ \\

$\Delta_{7.4}$ & $\Gamma_{8.50-51}$ &
$\Delta_{7.32}$ & $\Gamma_{8.145-149}$ &
$\Gamma_{7.54}$ & $\Gamma_{8.97,105,109,118,121,123}$ \\

$\Delta_{7.5}$ & $\Gamma_{8.52-53}$ &
$\Delta_{7.33}$ & $\Gamma_{8.159-160}$ &
$\Gamma_{7.55}$ & $\Gamma_{8.98,106,110-111,120,122-123}$ \\

$\Delta_{7.6}$ & $\Gamma_{8.54-56}$ &
$\Delta_{7.34}$ & $\Gamma_{8.154-156,163}$ &
$\Gamma_{7.56}$ & $\Gamma_{8.104,110}$ \\

$\Delta_{7.7}$ & $\Gamma_{8.58}$ &
$\Delta_{7.35}$ & $\Gamma_{8.157-158,164}$ &
& \\
\hline
\end{tabular}

\subsection{Antipodal quotients}\label{aq}
If, for some graph $\Gamma$, being at maximal distance is an
equivalence relation, we can define the quotient $\Gamma/\alpha$ of
$\Gamma$ under the map $\alpha$ sending each vertex to its equivalence class.
For $(0,2)$-graphs of valency 8, this produces a $(0,2)$-graph again
in four cases: the antipodal quotients of the four graphs
$\Delta_{8.13}$, $\Delta_{8.100}$, $\Delta_{8.104}$ and $\Gamma_{8.186}$
are the four graphs $\Gamma_{8.9}$, $\Gamma_{8.171}$, $\Delta_{8.71}$
and $\Gamma_{8.35}$, respectively.

\subsection{$\Gamma_{8.98}$ as quotient of a folded cube}\label{fq}
The graph $\Gamma_{8.98}$ is obtained from $2^8 / {\bf 1}$ by taking the
quotient under $x \mapsto \pi(x)+u$, where $\pi$ is the coordinate
permutation $(12)$, and $u = (00000111)$.

\subsection{$\Delta_{8.19}$ as cover of a directed graph}\label{dircov}
Let $\Sigma$ be a directed graph with vertex set $X$ of size $v$.
Construct an undirected cover $\Gamma$ with vertex set $\Zz_4 \times X$
of size $4v$ by letting $(i,x) \adj (i+1,y)$ when $(x,y)$ is an edge of
$\Sigma$. Let $T$ be the underlying undirected graph of $\Sigma$.
If in $T$ any two vertices are joined by zero or two paths of length 2
with both corresponding $\Sigma$-edges in the same direction,
and by zero or two paths of length 2 with the corresponding $\Sigma$-edges
in opposite directions, then each connected component of $\Gamma$
will be a (0,2)-graph. If $T$ is connected, then the graph $\Gamma$
is connected when $T$ is not bipartite, and has two connected components
otherwise.

In particular, consider the graph $\Sigma$ on the flags of the Fano plane,
with a directed edge $(p,L) \rightarrow (q,M)$ whenever $p \ne q$,
$L \ne M$ and $q \in L$. (Now $T$ is the distance-2 graph of the
generalized hexagon $GH(2,1)$.) It satisfies the given conditions,
and the graph $\Gamma$ is a (0,2)-graph on 84 vertices. It is
$\Delta_{8.19}$. The group is $4.L_3(2).2$, acting flag-transitively.
% with pt stab D_{16}

\subsection{$\Delta_{8.62}$ as cover of the co-Heawood graph}\label{l32}
Make the seven points of the Fano plane together with an additional
element 0 into a binary vector space $V$ of dimension 3,
by agreeing that the sum of three distinct collinear points is $0$.
Similarly, let $W$ be the binary vector space of dimension 3 spanned
by the lines, where the sum of three concurrent lines is $0$.
Construct a bipartite graph with 56 vertices $(p,w)$, where $p$ is a point
and $w \in W$, and 56 vertices $(L,v)$, where $L$ is a line and $v \in V$.
Let $(p,0) \adj (L,0)$ when $p \notin L$, and let $(p,M) \adj (L,q)$
when $p \notin L$ and $p,q \in M$ and $q \in L,M$, and finally,
whenever $(p,w) \adj (L,v)$, let also $(p,w+L) \adj (L,v)$
and $(p,w) \adj (L,v+p)$. The resulting graph is the (0,2)-graph
$\Delta_{8.62}$. It has automorphism group $2^6 . L_3(2) . 2$.
Mapping $(p,w)$ to $p$ and $(L,v)$ to $L$, we see that this graph
is an 8-cover of the co-Heawood graph.

\subsection{$\Gamma_{8.9}$ from $GQ(3,3)$}\label{gq}
Up to duality there is a unique generalized quadrangle of order 3.
It has 40 points and 40 lines.
One construction is given by the isotropic points and totally isotropic
lines in $Sp(4,3)$ (it has regular points, but lines are not regular).
The dual quadrangle is obtained by taking the isotropic points
and totally isotropic lines on $O(5,3)$.
A third construction, isomorphic with the first, is found by taking
the 40 nonisotropic points in $U(4,2)$, adjacent when orthogonal
(so that lines are orthogonal bases, and lines joining nonadjacent points
are tangents). Take the collinearity graph of this last generalized
quadrangle, fix a totally isotropic line $L$, and remove the edges
of the five $K_{4,4}$ subgraphs in the five tangent planes $x^\perp$
for $x \in L$. (In other words, remove an edge if the line carrying it
meets the line $L$.) The result is the (0,2)-graph $\Gamma_{8.9}$.
Its group is $2^4 . {\rm Sym(5)}$. Its bipartite double is $\Delta_{8.13}$.

\section{Acknowledgment}
Each of Aart Blokhuis, Koen Thas and Henny Wilbrink contributed to
the description of $\Gamma_{8.9}$ given in \S\ref{gq}.

\begin{thebibliography}{}

\bibitem{B} Andries E. Brouwer,
{\it Classification of small (0,2)-graphs},
J. Combinatorial Th. (A) {\bf 113} (2006) 1636--1645.

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

\bibitem{Hu} D. R. Hughes,
{\it Biplanes and semi-biplanes},
in: Combinatorial mathematics (Proc. Internat. Conf. Combinatorial Theory,
Australian Nat. Univ., Canberra, 1977), pp. 55--58,
Lecture Notes in Math. 686, Springer, Berlin, 1978.

\bibitem{JT} Zvonimir Janko \& Tran Van Trung,
{\it Two new semibiplanes},
J. Combin. Theory Ser. A {\bf 33} (1982) 102--105.

\bibitem{KO} P. Kaski and P. R. J. \"Osterg{\aa}rd,
  \emph{Classification Algorithms for Codes and Designs},
  Springer, Berlin, 2006.

\bibitem{K} D. E. Knuth, Dancing links, in \emph{Millennial 
  Perspectives in Computer Science}, J. Davies, B. Roscoe, 
  and J. Woodcock (Eds.), Palgrave, Basingstoke, England, 
  2000, pp.~187--214.

\bibitem{Mc} B. D. McKay, \emph{nauty} user's guide 
  (version 1.5), Technical Report TR-CS-90-02, 
  Computer Science Department, Australian National 
  University, Canberra, 1990.

\bibitem{Mo} Michel Mollard,
{\it Table of (0,2)-graphs of order less than 32},
Rapport de Recherche/IMAG RR 939, Grenoble, June 1994.

\bibitem{Mh} G. Eric Moorhouse,
{\it Reconstructing projective planes from semibiplanes},
in: Coding theory and design theory, Part II, pp. 280--285,
IMA Vol. Math. Appl. 21, Springer, New York, 1990.

\bibitem{Mu} Martyn Mulder,
{\it $(0,\lambda)$-graphs and $n$-cubes},
Discrete Math. {\bf 28} (1979) 179--188.

\bibitem{N} A. Neumaier,
{\it Rectagraphs, diagrams, and Suzuki's sporadic simple group},
in: Algebraic and geometric combinatorics, pp. 305--318,
North-Holland Math. Stud. 65, North-Holland, Amsterdam, 1982.

\bibitem{Wt} P. R. Wild,
{\it On semibiplanes}, Ph.D. Thesis, University of London, 1980.

\bibitem{W} Peter Wild,
{\it Generalized Hussain graphs and semibiplanes with $k\leq 6$},
Ars Combin. {\bf 14} (1982) 147--167.

\end{thebibliography}

\end{document}

Coulter, Robert S.(5-QLD-DMC); Henderson, Marie(5-QLD-DMC)
   A class of functions and their application in constructing semi-biplanes and
   association schemes. (English. English summary)
   Discrete Math. 202 (1999), no. 1-3, 21--31.

% What about nonbipartite quotients of a product?
