ADDITION.
On p. 258, line 6, add: `For a survey of the known geometries
that are extended grids, see
.SC Meixner & Pasini
.[[
Meixner Pasini 1993 extended grids
.]].'
CORRECTION.
On p. 258, line $-17$, and on p. 263, line 3,
replace `Corollary' by `Theorem'.
ADDITION.
On p. 258, after line $-6$, add:
.LP
On the other hand, there do exist regular graphs of diameter $d$, with
valency $k = d sup 2 + d + 1$ on $v = 2 Binom(2d,d)$ vertices satisfying
$c sub i = i sup 2$ ($1 <= i <= d$)
.SC (Brouwer & Koolen
.[[
Brouwer Koolen infinite series
.]],
cf. \(sc$Repres$.15).
Thus, the collection of Johnson graphs $J(n,d)$
is not characterized by $c sub i$ alone.
CORRECTION.
On p. 260, in Proposition 9.1.8, change the first part of the
third sentence to `$GAM$ is bipartite and antipodal;
if $m > 0$ it is the bipartite double of the Odd graph on $X$'.
ADDITION.
On p. 261, at the end of Section 9.1, add to Remark (iv):
.LP
Muzichuk's result appeared as
.[[
Muzychuk subschemes Johnson 1992
.]].
.SC Uchida
.[[
Uchida 1992
.]]
improved Muzichuk's result and showed that merging classes
in $J(n,m)$ does not lead to an association scheme for
$n >= f(m)$, where $f(3) = 11$, $f(4) = 13$, $f(5) = 15$,
$f(6) = 18$ and $f(m) = 3m-1$ for $m >= 7$.
Clearly, this is best possible for $m = 3,4$.
ADDITIONS.
On p. 261, at the end of Section 9.1, add a Remark (v) stating
that $2.O sub m$ is a geodetically closed induced subgraph of
$O sub 2m$ (with the same diameter $2m-1$).
Add a Remark (vi):
.SC Meyerowitz
.[[
Meyerowitz cycle-balanced
.]]
classified the completely regular designs of strength 0 in the Johnson
scheme.
.SC Martin
.[[
Martin completely regular strength one 1994
.]]
classified the completely regular designs of strength 1 and
minimum distance 2 in the Johnson scheme.
ADDITION.
On p. 262, in the remark following Corollary 9.2.3, add:
.LP
Several other people characterized the $d$-cube by its parameters
(i.e., by being bipartite of diameter $d$
with $c sub i = i$ for $1 <= i <= d$; note that a rectagraph is
always regular) - cf. e.g.
.SC Foldes
.[[
Foldes 1977 hypercubes
.]]
and
.SC Mulder
.[[
Mulder 1982 interval-regular
.]].
ADDITION.
On p. 263, the question in the Problem can be answered negatively:
.SC Haemers
.[[
Haemers distance regularity spectrum graphs 1992
.]]
has constructed non-distance-regular graphs
cospectral with $Binom(n,3)$, $n >= 6$.
(Moreover, Hoffman's graph described above arises from $2 sup 4$
by `Haemers switching'.)
ADDITION.
On p. 265, add Remark (ia): `The folded $(d+1)$-cube is
$GAM sub 1 cu GAM sub d$ of the $d$-cube.'
CORRECTION.
On p. 272, line 14-15, delete the sentence `It is ...
Grassmann graph $Qnom(V,m)$'.
ADDITION.
On p. 273, the problem has been answered completely.
Add at the end of Remark (iii):
.LP
Gardiner asked whether any distance-transitive graph with parameters
$q . K sub q,q$ must be derived from the Desarguesian projective plane,
but
.SC Chuvaeva & Pasechnik
.[[
Chuvaeva Pasechnik distance-transitive planes
.]]
showed that also certain planes coordinated by the twisted
fields of Albert gave rise to distance-transitive graphs, and
.Ax "Albert, A.A."
subsequently
.SC Liebler
.[[
Liebler classification distance-transitive type
.]]
showed that conversely any such distance-transitive graph is derived
from either the Desarguesian projective plane, or a plane
coordinatized by a twisted field of Albert.
(However, now Liebler says that his proof may be incomplete.)
CORRECTION.
On p. 279, in Theorem 9.4.10, insert after `(6)': `but with $m+1$
instead of $m$'.
CORRECTION.
On p. 284, lines 1-2: delete the statement about the group
in Theorem 9.5.6. (The precise result was already given in Theorem 9.5.3.)
ADDITION.
On p. 284, bottom, add: `(iv) A local characterization of
the alternating forms graphs was given by
.SC Munemasa & Shpectorov
.[[
Munemasa Shpectorov 1993 local characterization MSonly
.]]
and by
.SC Munemasa, Pasechnik & Shpectorov
.[[
Munemasa Pasechnik Shpectorov 1993 local characterization
.]].'
ADDITION.
On p. 285, in Theorem 9.5.7, the group of automorphisms given
is (for $d >= 2$) the full automorphism group of $GAM$
.SC (Ivanov & Shpectorov
.[[
Ivanov Shpectorov characterization association schemes Hermitian 1991
.]]).
In the same paper the authors show that
if $GAM$ is a distance-regular graph with parameters (7)
and singular lines of size $r$, and $d >= 3$, then $GAM$ is the Hermitean
forms graph on $Ff sub q supr(n)$. They remark that, according to a
personal communication by P. Terwilliger, the assumption of
singular lines of size $r$ is superfluous. Finally, they remark that
probably
.SC Wan
.[[
Wan 1965 Scientice Sinica
.]]
was the first to compute the parameters of the association scheme
of Hermitean forms.
CORRECTION & ADDITIONS.
On p. 286 the discussion following Lemma 9.5.9 is somewhat imprecise
in the case where $q$ is even, as was pointed out by Pasechnik.
The facts are as follows:
Let $GAM$ be the symmetric bilinear forms graph on $Ff sub q supr(n)$
where $q$ is even. Then $GAM$ is distance-regular for $n <= 3$ but not
for $n > 3$. The distance function is given by
$d sub GAM (f,g) = rk (f-g)$ when $f-g$ is not alternating, and
$d sub GAM (f,g) = rk (f-g) + 1$ otherwise.
Each nonalternating form $f$ of rank $i$ has $q sup i-1$ neighbours
of rank $i-1$, and $q sup n - q sup i$ neighbours of rank $i+1$, and
a unique neighbour (of rank $i-1$ or $i$) that is alternating.
Thus, the alternating forms form a perfect code in $GAM$.
Each alternating form $g$ of rank $i$ has $q sup i - 1$ neighbours
of the same rank.
For $n = 3$ we find a distance-regular graph with intersection array
$"{"q sup 3 -1 , q sup 3 -q , q sup 3 -q sup 2 + 1 ;~ 1,q,q sup 2 -1"}"$
(in the diagram on p. 286, merge the balloons for the forms of rank 3 and
the alternating forms of rank 2).
The distribution diagram for $n = 4$ is
.so diag/symbilin4
For even $q$, there is a 1-1 correspondence $phi$ between symmetric bilinear
forms on $Ff sub q supr(n)$ and alternating forms on $Ff sub q supr({n+1})$
given by
.EQ
( phi f )((x, xi ), (y, eta )) = f(x,y) + sqrt f(x,x) sqrt f(y,y) +
eta sqrt f(x,x) + xi sqrt f(y,y) .
.EN
(This remark may be due to Kantor.)
Under this correspondence, the symmetric bilinear forms of rank $2i-1$ or $2i$
correspond precisely to the alternating forms of rank $2i$.
(The geometric version of this correspondence follows by regarding the symmetric
bilinear forms as the vertices far away from a given vertex in $B sub d (q)$,
and the alternating forms as the vertices far away from a given vertex in
$D sub d+1 (q)$ where the $B sub d$-geometry lives on a nondegenerate hyperplane
in the $D sub d+1$-geometry (cf. the discussion on p. 278).)
[However, it does not follow that the distance 1-or-2 graph of $GAM$ is the
alternating forms graph, since not all symmetric bilinear forms of rank 2
lie in $GAM sub 2 (0)$.]
SPELLING CORRECTION.
On p. 287, line $-20$, replace `The' by `Then'.
CORRECTION.
On p. 289, line 20, replace `\(sc14' by `\(sc11.3H'.
ADDITION.
On p. 293, Remarks, add references to
.SC Brouwer, Hemmeter & Woldar
.[[
Brouwer Hemmeter Woldar complete list cliques
.]]
and
.SC Hemmeter & Woldar
.[[
Hemmeter Woldar complete list cliques evencase
.]].
Add the remark: `E.W. Lambeck remarks that when $q$ is even
the partition into $q sup n$-cliques defined by the equivalence
relation $gam == del$ if and only if $rk ( del - gam ) <= 1$
is completely regular (if $d( gam , C) = 1$ for such a clique $C$,
then $vb gam sup perp ca C vb = q sup 2$) and the corresponding
quotient is the alternating forms graph on $V$.'
.LP
Add the remark:
.SC `Munemasa, Pasechnik & Shpectorov
.[[
Munemasa Pasechnik Shpectorov 1993 automorphism convex
.]]
determine the full automorphism group of the quadratic forms graph
in the characteristic 2 case. Interestingly,
it is larger than expected. They also determine the convex subsets
of this graph, thus completing Lambeck's work.'