CORRECTION.
On p. 90-91, Theorem 3.6.1 and its proof were formulated somewhat
imprecisely. A polished version follows.
.LP
.B
3.6.1 Theorem.
.I
Let $GAM$ be a graph on $v$ vertices
with a $(p,q,r)$-representation in $Rr sup n$, $p != q,r$.
Then $GAM$ has at most $1o2 n(n + 3)$ vertices.
Moreover, if $GAM$ is a disjoint union of cliques, then
$v <= n + 1$, except when some component of $GAM$ has size
$s := (p-q) slash (r-q)$ (so that $s$ is an integer $X> 1$);
in this case $v <= n + t - del <= 2(n - del )$, where $t$ is the number
of maximal cliques of size $s$, and $del = 1$ if $r != 0$, $del = 0$
if $r = 0$.
.Bop
\&...
Since $p != q$, a straightforward calculation
shows that $rk G bar >= r - 1$ unless $GAM$ has components of size
$s = (p-q) slash (r-q)$;
in this case $rk G bar = r + del - t$, where $t$ is the
number of components of size $s$. Now ...
.Eop
[The text of the proof is ugly as well; the variable $r$ occurs
in two r\o'o'les; call the number of cliques $m$ instead of $r$.
Also, in line 7 on p. 91, replace `matrix $G$' by `space $Rr sup n$'.]
SIMPLIFICATION.
On p. 92-93, change the proof of Proposition 3.7.3 into:
.LP
.B Proof.
Let $alpha bar$ denote the image of a vertex $alpha$ under
the given representation. Now define a new representation
by projection onto ${ gam bar } sup perp$, i.e.,
by sending (in case (i)) $del mo GAM ( gam )$ to
$del bar - smover(q,p) gam bar$, and (in cases (ii), (iii))
$del nm gam sup perp$ to $del bar - smover(r,p) gam bar$.
.Eop
CORRECTION.
On p. 104, line 7, the parameters should be (16,5,0,2).
Similarly, in the index on p. 480, line 11, this parameter
set should be listed as {5,4; 1,2}.
CORRECTION.
On p. 106, line $-10$, the reference should have been to
.SC Hoffman
.[[
Hoffman graphs whose least eigenvalue exceeds 1977
.]].
ADDITION.
On p. 110, before Theorem 3.12.5, add a reference to
.SC Beineke
.[[
Beineke derived 1970
.]]
for characterizations of line graphs by forbidden subgraphs.
ADDITION.
On p. 111, after line 5, add a reference to
.SC Woo & Neumaier
.[[
Woo Neumaier smallest eigenvalue
.]]
for a characterization of graphs with large minimum valency
and smallest eigenvalue at lest $- 1 - sqrt 2$.
CORRECTION.
On p. 114, line 3, change `3.10' into `3.11'.
ADDITION.
On p. 114 add to the list of examples of amply regular code graphs
the polygons, the dodecahedron and the doubled Odd graphs.
CORRECTION.
On p. 116 the conclusion of Theorem 3.15.1 in case $mu = 4$
is incomplete: there is another family of regular root graphs
with $mu = 4$. See
.SC Brouwer & Koolen
.[[
brouwer koolen geodetic
.]].
CORRECTION.
On p. 120 the proof of Proposition 3.15.2 is incomplete
in case $mu = 2$, $lam > 0$, since it is not immediately clear
why $GAM$ should be locally connected. However, if $GAM ( gam )$
is disconnected, then it must contain a $( lam + 2)$-clique,
and it is not difficult to check that this clique will be
a direct factor of $GAM$. But then $Ll ( GAM )$ is decomposable,
contradiction.
CORRECTION.
On p. 125 the proof in Case 3 is incorrect.
Insert in the line before Case 1 a new condition S9,
and change the text in Case 3 as follows.
.IP S9.
If $DEL sub 0$ strictly contains an $n$-gon, then $DEL sub 0$ has
girth 5, and $n = 5$ or $3 vb n$.
.LP
Indeed, if $P$ is an $n$-gon in $DEL sub 0$ with $n >= 6$ and
$a$ a vertex at distance 1 to $P$, then S8 implies that $a$ is adjacent
to every third vertex of $P$, so that $3 vb n$.
.SC Case 3.
.I
$DELTA sub 0$ has girth $5$.
.R
Then $DELTA sub 0$ is not bipartite, and since $L( DELTA sub 0 )$
must be regular of valency $lambda$, $DELTA sub 0$ is regular
of valency $1o2 lambda + 1$. If $lambda = 2$ then $DELTA sub 0$
is the pentagon and $GAM$ is locally a disjoint union of pentagons.
Now the set of all $del mo GAM$ such that all edges in $S( del )$
lie in $DEL sub 0$ is easily seen to be an icosahedron,
and by our assumptions $GAM$ itself is an icosahedron.
If $lambda > 2$, then
for every vertex $i$ of a pentagon $1 adj 2 adj 3 adj 4 adj 5 adj 1$
of $DELTA sub 0$, the set $N (i)$ of neighbours
of $i$ outside this pentagon is a $( 1o2 lambda - 1 )$-coclique, and by S8,
$N(i) cu N(i+2)$ ($i ~ mod 5$)
is a complete bipartite graph. Since there are no quadrangles,
this forces $lambda = 4$, and $DELTA sub 0$ is the Petersen graph,
so that $GAM$ is locally a disjoint union of line graphs of the
Petersen graph.
Now every $K sub 1 + K sub 1,2$
of the Petersen graph is in some path of length 4, and S7 shows
that every 4-set containing two intersecting edges of $DELTA sub 0$
is special. But there are 8 such 4-sets through an edge $ab$
of $DELTA sub 0$, and they determine 8 neighbours of the point $del$
of $GAMMA$ represented by $ab$, forming an 8-gon in $GAM ( del )$.
But $3 notdiv 8$, contradicting S9.
.Eop