CORRECTION.
On p. 195 in Table 6.2, change `9.4.5' into `9.4.6, 9.4C'.
CORRECTION.
On p. 195, the statement and proof of Proposition 6.1.2 are incorrect.
.LP
.B
6.1.2. Proposition.
.I
The intersection numbers of every distance-regular graph
of diameter $d >= 3$ with classical parameters satisfy
.EQ C
c sub i+1 > ci ~~~~ roman for ~~ i = 2 , ... , d-1 ;
.EN
in particular the geometric girth of such a graph cannot exceed $6$.
.Bop
\&...
therefore $Qn(i) = \(+- 1$ which forces $i = 1$.
.Eop
Note that the generalized hexagons have geometric girth 6.
CORRECTION.
On p. 200-201 the (trivial) case of generalized 2-gons is not handled
correctly. On p. 200 line 4 of \(sc6.5, replace `partial linear space'
by `point-line geometry'; on line $-13$ replace `linear space'
by `a geometry'; on line $-12$ replace `the partial linear space'
by `it'; on the last line of p. 200 and the first line of p. 201
insert `2,'.
ADDITION.
On p. 204, add: `There are unique generalized quadrangles of
orders $(s,1)$, $(1,t)$, $(2,4)$, $(4,2)$, $(3,5)$, $(5,3)$,
$(3,9)$ and $(9,3)$. No generalized quadrangle of order $(3,6)$ exists.
For orders $(3,5)$, $(3,6)$ and $(3,9)$, see
.SC Dixmier & Zara
.[[
Dixmier Zara etude autour 1976
.]]
and
.SC Payne & Thas
.[[
Payne Thas generalized quadrangles Pitman
.]].'
.KS
CORRECTION.
On p. 205, something went wrong in the last three
lines of Table 6.6. These should have been:
.TS
expand,center,delim($$),tab(@);
l1 l1 l1 c1 c1 c1 l.
_
@name@\(sc@$n$@$k$@$lambda$@$c sub i$
_
(R4)@Folded cubes (P1)@9.2D@$n$@$n$@0@$ c sub i = i (i < d), c sub d = gamma d$
(R5)@Hamming graphs (C8)@9.2@$2d$@$sd$@$s-1$@$c sub i = i$
(R6)@Dual polar graphs (C3)@9.4@$2d$@$q sup e Qn(d) sub q$@$q sup e -1$@$c sub i = Qnomq(i,1)$
(R6a)@Pseudo $D sub m (q)$ graphs (C3a)@9.4.6, 9.4C@$2d$@$Qn(d) sub q$@0@$c sub i = Qnomq(i,1)$
_
.TE
.KE
CORRECTION.
On p. 206, in Table 6.7, line (N2), the group should be
$3 sup 6 . 2 . M sub 12$.
In the statement of Theorem 6.6.1, replace `$mu = 1$' by `$mu <= 2$'.
In the subsequent discussion replace `$mu > 1$' by `$mu > 2$'.
ADDITION.
On p. 206, at the end of Section 6.6, add:
.LP
Mathon (pers. comm.) showed that the Krein condition
$q sub 33 sup 2 >= 0$ for regular near hexagons is equivalent
to $t <= s sup 3 + ( mu - 1)(s sup 2 - s + 1)$ if $s > 1$.
(For this, and more information on Krein conditions for near polygons,
cf.
.SC Brouwer & Wilbrink
.[[
Brouwer Wilbrink 1983 structure near polygons
.]].)
A generalization was given in
.SC Neumaier
.[[
Neumaier 1990 Krein conditions
.]].
ADDITION.
On p. 207, add after line 10:
.SC `Cameron
.[[
Cameron Beineke Wilson 1983
.]],
Proposition 5.4, shows that no example with $k = 57$ can be vertex-transitive,
.Ax "Higman, G."
a result due to G. Higman (cf. Proposition 2.2.9).'
ADDITION.
On p. 207, add in line 18:
.SC `J\*(/orgensen
.[[
J orgensen diameters cubic graphs 1992
.]]
shows that no cubic graph of diameter $d >= 4$ has defect 2.'
ADDITION.
On p. 208, replace lines $-4$, $-3$ by:
.LP
and
.SC Roos, van Zanten & Coster
.[[
Coster Roos van Zanten existence Moore 1986 part4
.]]
and
.SC Damerell, Roos & van Zanten
.[[
Damerell Roos van Zanten certain Moore 1989
.]],
where it is shown that the eigenvalues are at most quadratic,
and that this array is not feasible for $d >= 4$ when $s t > 1$).
ADDITION.
On p. 209, line 1, insert:
.LP
.SC Van Zanten
.[[
van Zanten 1989 short proof
.]]
gives another proof of the consequence of $Classif$.8.2 (ii) that
all eigenvalues of $GAM$ are at most quadratic over $Qq$.
CORRECTION.
On p.\|210, in the description of the Robertson-Wegner graph
it is stated that the full group has order 20 with three point orbits.
Gordon Royle pointed out that in fact the full group has order 120
and has two point orbits (of sizes 10, 20). Of course this means
that there is a much nicer description, and indeed there is:
The 30 vertices are the 20 vertices of the dodecahedron and the
10 4-subsets of the dodecahedron that have all internal distances 3;
the adjacencies are the obvious ones: the dodecahedron is an induced
subgraph of valency 3, each 4-subset is adjacent to its 4 elements
and to the antipodal 4-subset.
This description shows the (full) group $2 times Alt(5)$.
It is also stated that the subgraph induced by 3 pentagons and
3 pentagrams in the Hoffman-Singleton graph has automorphism group
of order 10. Gordon Royle points out that the order is 20 (with
orbit sizes {5,5,10,10}).
Part of the confusion is caused by the fact that the pictures drawn
in Wong [cages, a survey] do not correspond to the descriptions given.
Quoting Royle [email, 951222]: the Robertson-Wegner graph is Wong's
figure 4 (not 5); the 3 pentagons + 3 pentagraphs graph is Wong's
figure 5 (not 4). Wong's figure 6 is a (5,5)-cage constructed by
Foster, with group of order 30 and orbit sizes {15,15}.
It is rumoured that there is precisely one further (5,5)-cage.
CORRECTION.
On p. 210, in the discussion of the (7,6)-cage, change `Apparently ...
$L sub 2 (7)$' into `However, as Baker [pers. comm.] (see also
.SC Baker
.[[
Baker automorphisms elliptic 1983
.]],
p. 96) pointed out, the elliptic semiplane is self-dual, and
in fact $Aut GAM isom 3 cdot Alt (7) . 2$, acting transitively
on $GAM$.
($Aut GAM$ has rank 8; the two antipodes of a vertex $gam$
cannot be interchanged by an automorphism fixing $gam$.)'
ADDITION.
On p. 210, at the end of Section 6.9, add: `,
.SC Biggs & Boshier
.[[
Biggs Boshier 1990 girth Ramanujan
.]],
.SC Lazebnik & Ustimenko
.[[
Lazebnik Ustimenko 1993
.]]'.
Six lines earlier, after the reference to
.SC Wong
[793], add: `An older reference is
.SC Walther & Voss
.[[
Walther Voss Kreise 1974
.]],
Teil II, Kap. 1'.
ADDITION.
On p. 213, after `are determined.' add: `(The arguments were
geometric in nature. A treatment using parameters only was
given by
.SC Terwilliger
.[[
Terwilliger 1993 association schemes covers
.]].'
ADDITION.
On p. 225, bottom, add: `The antipodal distance-transitive covers of
$K_k,k$ have been classified by
.SC Ivanov, Liebler, Penttila & Praeger
.[[
Ivanov Liebler Penttila Praeger antipodal
.]].'
ADDITION.
On p. 229, at the end of \(sc7.6, add: `The 2-transitive
two-graphs and their full groups of automorphisms
have been determined by
.SC Taylor
.[[
Taylor two-graphs doubly transitive groups
.]];
there are no surprises. In particular, in cases (v) and (vi), we have
$Aut GAM isom 2 times PGAMU (3,q sup 2 )$ and
$2 times "" sup 2 G sub 2 (q) . Aut Ff sub q$, respectively,
except for $q = 3$ where $Aut GAM isom 2 times Sp (6,2)$.'