CORRECTION.
On p. 380, line 13, middle, replace `$p sub 23 sup 3 = q-3$'
by `$p sub 23 sup 2 = q-3$'.
ADDITION.
On p. 382, after the fourth line of \(sc12.3, add:
.LP
.SC Coxeter
.[[
Coxeter systematic notation 1981
.]]
writes ``This `Coxeter graph' was discovered independently by
J.H. Conway and R.M. Foster''.
.LP
Before Theorem 12.3.1, add: `See also
.SC Coxeter
.[[
Coxeter systematic notation 1981
.]]
for two drawings'.
ADDITION.
After p. 384, add a new section 12.4A.
.Na "A" "The Meixner graphs"
It was pointed out to us by L. Soicher, that
.SC Meixner
.[[
Meixner polar towers
.]]
(especially Proposition 4.3)
implicitly contains the following.
Let $DEL$ be the graph on the nonisotropic points in $U(6,2 sup 2 )$,
where two points are adjacent when joined by a tangent.
The universal cover $DEL hat$ of $DEL$ (among the covers that are
locally isomorphisms) is a distance-transitive antipodal 4-cover
with parameters $"{"176,135,36,1;1,12,135,176"}"$ and
automorphism group $2 sup 2 . U(6,2) . Sym (3)$.
The graph $DEL$ also has a distance-transitive antipodal 2-cover
with parameters $"{"176,135,24,1;1,24,135,176"}"$ and
automorphism group $2 . U(6,2) . 2$.
ADDITION.
On p. 386, add after line 11: `The graphs with intersection array
$"{"8,6,1;1,3,8"}"$ were determined again by
.SC Juri\*Vsi\*'c
.[[
Juri distance regular covers 1991
.]].
Add after line 14: `Generalized quadrangles of order $(q sup 2 ,q)$
do not have spreads, and no $GQ(6,3)$ exists. In cases where it is
possible to establish the existence of lines, this can be used to
rule out such distance-regular graphs (cf.
.SC Godsil
.[[
Godsil geometric distance-regular covers
.]]).
For example, using the remark following Proposition 1.2.1 it follows
that there are no graphs with intersection array
$"{"18,12,1;1,2,18"}"$ $(v = 133)$ or
$"{"27,18,1;1,2,27"}"$ $(v = 280)$.'
ADDITION.
On p. 386, add to Remark (iii) that it follows from
.SC Lenstra
.[[
Lenstra automorphisms finite fields 1990
.]]
that $Aut GAM$ is no larger than the group described.
ADDITION.
On p. 386, last line, add: `This graph is known as the Klein graph.
It is locally a heptagon. See also
.SC Curtis
.[[
Curtis geometric interpretations Mathieu 1990
.]]
and Coolsaet
.[[
Coolsaet local structure
.]].'
Add entries `Klein graph, 386' and `locally heptagon, 386'
to the Subject index on pp. 491, 492.
ADDITION.
On p. 387, at the end of \(sc12.5, add:
.LP
In a completely analogous way
.SC Cameron
.[[
Cameron covers EGQs
.]]
constructs covers of complete graphs using Hermitean or quadratic
forms. Let us look at the Hermitean case first.
[Case (i) below was not given by Cameron.]
.LP
.B
12.5.4. Proposition.
.I
Let $q sup 2 = m r + 1$ for some prime power $q$, and let $V$
be a vector space of dimension 3 over $F = Ff sub { q sup 2 }$
provided with a nondegenerate Hermitean form $H$.
Let $K$ be the subgroup of index $r$ of the multiplicative group $F star$,
and choose $b mo F star$ such that $b K = b bar K$ (i.e., such that
$b sup q-1 mo K$).
Define a graph $GAM$ with vertex set $"{" K v vb v mo V minus "{"0"}" "}"$
where $K u adj K v$ if and only if $H(u,v) mo b K$.
Then $GAM$ is an antipodal $r$-cover of the complete graph
$K sub { q sup 3 + 1 }$. The graph $GAM$ is distance-regular
if and only if one of the following three conditions hold.
.IP (i)
.I
$r$ is odd and $r vb (q-1)$,
.IP (ii)
.I
$q$ is even and $r vb (q+1)$,
.IP (iii)
.I
$q$ is odd and $r vb 1o2 (q+1)$.
.LP
.I
In case (i) we have $lam = mu = (q sup 3 - 1)/r$;
in cases (ii) and (iii) we find
$lam = (q+1)m - q(q-1)$ and $mu = (q+1)m$.
.LP
.B Proof.
Everything is clear except for distance-regularity.
(The vertices $K u$ contained in the same projective point
$F u$ are antipodes: they have mutual distance at least 3,
and in fact precisely three, as we shall see soon.)
The group $GU (3,q sup 2 )$ acts transitively on the vertices
and edges, and 2-transitively on the projective points $F u$
($u mo V minus "{"0"}"$). For distance-regularity it is
necessary and sufficient that $mu$ is constant. Thus,
let $K u$ and $K v$ be two vertices not contained in the same
projective point, and choose coordinates such that ...
.Eop
.LP
.B
12.5.5. Proposition
.R
.SC (de Caen, Mathon & Moorhouse
.[[
de caen Mathon Moorhouse Preparata
.]]).
.I
Let $q = 2 sup 2t-1$ and $s = 2 sup e$, where $gcd(e,2t-1) = 1$.
Construct a graph $GAM$ whose vertices are the triples $(a,i, alpha )$
with $a, alpha mo Ff sub q$ and $i mo Ff sub 2$,
and in which $(a,i, alpha ) adj (b,j, beta )$ when $alpha + beta =
a sup s b + a b sup s + (i+j)(a sup s+1 + b sup s+1 )$.
Then $GAM$ is distance-regular with intersection array
$"{"2q-1,2q-2,1;~1,2,2q-1"}"$.
Taking the quotient w.r.t. $"{"0"}" times "{"0"}" times A$,
where $A$ is a subgroup of the additive group of $Ff sub q$,
yields a distance-regular graph $GAM bar$ with intersection array
$"{"2q-1,2q-2 sup i ,1;~1,2 sup i ,2q-1"}"$
when $vb A vb = 2 sup i-1$ ($1 <= i <= 2t-1$).
.LP
ADDITION.
On p. 387, after the description of the Denniston arcs, add: `; for
a characterization, cf.
.SC Abatangelo & Larato
.[[
Abatangelo Larato 1989 Denniston
.]]'.
CORRECTION.
On p. 389, Lemma 12.7.4, line 8, replace `if and only if' by `if'
and insert the following text after the lemma.
.LP
Mathon [loc. cit., p. 136] stated that the above cubic polynomial
in $theta$ is reducible if and only if $b = c$, but as Bannai
.Ax "Mathon, R.A."
.Ax "Bannai, Eiichi"
.Ax "Munemasa, A."
pointed out that is false. Munemasa gives the counter\%example
$a = 36$, $b = 42$, $c = 36$ corresponding to a scheme
$Zz sub { 7 sup 3 } . Zz sub 114$ on $v = 7 sup 3$ letters
(the cyclotomic scheme with $q = 7 sup 3$, $r = 3$, see below).
Here the $theta sub i$ are 9, 2, $-12$.
ADDITION.
On p. 389, in the remark after Proposition 12.7.5,
add a reference to
.SC Lenstra
.[[
Lenstra automorphisms finite fields 1990
.]].
ADDITION.
On p. 390, at the end of Chapter 13, add a new Section 12.8.
.nr H1 12
.nr H2 7
.Nh "The Pasechnik graphs and related graphs"
Let $V$ be a vector space of dimension 3 over the field $Ff sub q$,
provided with an outer product $times$.
Let $GAM$ be the graph with vertex set $V times V$ where
$(u,u prime ) adj (v,v prime )$ if and only if
$(u,u prime ) != (v,v prime )$ and $u times v + u prime - v prime = 0$.
Then $GAM$ is distance-regular with intersection array
$"{"q sup 3 - 1, q sup 3 - q, q sup 3 - q sup 2 + 1 ;~ hyph
1, q, q sup 2 - 1 "}"$.
Its extended bipartite double is distance-regular with intersection array
$"{"q sup 3 , q sup 3 - 1, q sup 3 - q, q sup 3 - q sup 2 + 1;~ hyph
1, q, q sup 2 - 1, q sup 3 "}"$ and was first constructed by
D. Pasechnik [email, March 1991].
When $q$ is a power of two, these graphs have the same parameters as
certain Kasami graphs, but for $q > 2$ they are nonisomorphic.