ADDITION.
On p. 135, at the end of \(sc4.1D, add:
.SC `Lambeck
.[[
Lambeck elementary inequalities 1993
.]]
observes that the specialization
$a sub d <= p sub d,d-1 sup d + p sub d,d sup d$
whenever $p sub d,d sup d != 0$ rules out the intersection array
$"{"39,32,20,2;1,4,16,30"}"$ $(v = 768)$.'
ADDITION.
On p. 136, line $-11$, insert the sentence:
.SC `Paulus
.[[
Paulus conference 1973
.]]
found conference graphs on 26 vertices with trivial group of
automorphisms.'
CORRECTION.
On p. 136, part (ii) of Theorem 4.1.10 is incorrect.
In fact the stabilizer of a single antipodal class
is 2-transitive on that class, but need not stabilize
any other classes, or be normal in $G$.
On the other hand, the stabilizer $N$ of all antipodal
classes is normal in $G$, but may be reduced to the identity.
ADDITION.
Delorme's preprint has now appeared. On p. 138, add a reference to
.SC Delorme
.[[
Delorme biregular 1994
.]].
CORRECTION.
On p. 138, line $-2$, replace `and $c sub i+1 Prime > c sub i Prime$'
by `and $c sub i+1 Prime > c sub i Prime2$'.
ADDITION.
On p. 139 add before the remarks, after `$d > 4$.':
.LP
For $d = 4$ there is a sporadic
example on $100 + 280$ vertices belonging to the Hall-Janko group
(cf.
.SC Bagchi
.[[
Bagchi regular two-graph admitting Hall-Janko-Wales
.]]).
Add at the end of \(sc4.2G:
.LP
(iii)
.SC Nomura
.[[
Nomura intersection diagrams biregular
.]]
obtained some parameter conditions using distance distribution diagrams
around an edge.
.LP
(iv)
.SC Suzuki
.[[
Suzuki distance-biregular divisible four
.]]
classifies the distance-biregular graphs of girth divisible by 4
with some trivalent vertices.
Add at the end of \(sc4.2H a reference to
.SC Hilano
.[[
Hilano 1989
.]].
ADDITION.
On p. 143, Corollary 4.2.6 can be sharpened somewhat
(following Godsil & Hensel).
.LP
.B
4.2.6. Corollary.
.I
In particular, in case $d = 1$, $d tilde = 3$, the two new eigenvalues
$theta$ are the two roots of $theta sup 2 - (k - 1 - r mu ) theta - k = 0$,
(with $mu = mu ( GAM tilde )$), and have multiplicity
$m( theta ) = { k(k + 1)(r - 1) } over { k + theta sup 2 }$.
Moreover, $k mu$ is even, and if $k != r mu + 1$, then $theta$
is an integer.
.Bop
We find that $(r - 1) k mu$ is even by counting edges in
$GAM tilde ( inf )$ for some $inf mo GAM tilde$.
Suppose $r == k == mu == 1$ (mod 2). Then the eigenvalues $theta$
are odd integers, and their sum is also odd, contradiction.
.Eop
Almost all of the above is already contained in
.SC Biggs & Gardiner
.[[
Biggs Gardiner classification
.]];
the parity condition in 4.2.6 is from
.SC Godsil & Hensel
.[[
Godsil Hensel covers 1989
.]].
.SC Godsil, Juri\*Vsi\*'c & Schade
.[[
Godsil Juri Schade antipodal strongly 1990
.]]
discuss the case $d tilde mo "{"4,5"}"$,
and in particular find that when $d tilde = 5$
at least one of the three new eigenvalues is integral.
ADDITION.
On p. 145, after Proposition 4.2.11 add:
.SC `Juri\*(Vsi\*('c
.[[
Juri\*(Vsi\*('c antipodal covers strongly regular graphs 1992 preprint
.]]
notes that if $E$ is any distance-regular graph that is both
bipartite and antipodal and has odd diameter, then it is a
2-cover (by the array given on p. 142) and hence 4.2.11 applies.
ADDITION.
On p. 146, line 2, add: `The Odd graphs $O sub d+1$
are the only generalized Odd graphs with $c sub 2 = 1$,
$c sub 3 = c sub 4 = 2$ [Koolen].'
SPELLING CORRECTION.
On p. 146, line $-4$, replace `the' by `then'.
ADDITION.
On p. 147, bottom, add:
.SC `Suzuki
.[[
Suzuki note association schemes polynomial structures 1993
.]]
shows that case (iii) of the above proposition does not occur at all.'
LAYOUT CORRECTION.
On p. 149 the page heading lost its white space.
CORRECTION.
On p. 151, line 6, the intersection array should be
{231,160,6; 1,48,210}.
ADDITION.
On p. 151, after the Problem, add:
.LP
(This question was posed for bipartite $GAM$ by
.SC Vetch\*'y
.[[
Vetch\*('y metrically regular square 1990
.]]
who gave partial results. But from the above and Corollary $Param$.4.2
one sees that the only such bipartite $GAM$ with $d >= 4$
are those of examples (i) and (ii) above; for $d = 3$ one finds
precisely the graphs ${ 2 times m } bar$.)
ADDITION.
On p. 151, at the end of \(sc4.2, add:
.LP
Nowadays an association scheme obtained by merging classes
is usually called a \fIfusion scheme\fP, cf. \(sc2.4A.
ADDITION.
On p. 152, part (iv) of Proposition 4.3.2 may be deleted, since
this condition is never fulfilled (E.W. Lambeck, cf. an addition
to p. 174). Partly generalizing Proposition 5.4.3 we have
a (rather weak) sufficient condition for the existence of singular
lines:
.IP (v)
.I
$c sub i-1 + b sub i > b sub 1$
.LP
.B Proof.
(In case (v).) Consider a triangle $alpha beta gamma$.
If there exists a vertex $del mo GAM sub 2 ( alpha ) ca
beta sup perp ca gam sup perp$ then there are
$p sub i-1,i+1 sup 2 + p sub i-2,i sup 2$ vertices $eps$
with distances $i-1,i,i,i+1$ or $i,i-1,i-1,i-2$ to
$alpha , beta , gam , del$, respectively.
On the other hand, there are $p sub i,i-1 sup 1$ vertices with
distances $i-1,i,i$ or $i-1,i,i-1$ to $alpha , beta , gam$
and the number of vertices with distances $i-1,i,i-1$
equals that with distances $i,i-1,i-1$ to $alpha , beta , gam$
(both counts equal $p sub i-1,i sup 1$ minus the number
of vertices with distances $i,i,i-1$),
so we find $p sub i-1,i+1 sup 2 + p sub i-2,i sup 2 <=
p sub i,i-1 sup 1$, i.e., $b sub i + c sub i-1 <= b sub 1$. $eop$
ADDITION.
On p. 152, add a Proposition 4.3.5 stating that if some singular
line in $GAM$ has size $lam + 2$, then $lam + 2 <= 1 + k/(- theta sub d )$
(because of 4.4.6 (i)). This rules out the intersection array
$"{"45,40,11,1;1,1,40,45"}"$ $(v = 2352, theta sub 4 = -11)$.
.LP
Add a subsection:
.LP
.B "Big cliques"
.LP
As
.SC Godsil
.[[
Godsil geometric distance-regular covers
.]]
remarked, the standard claw-and-clique arguments due to Bose
can sometimes be applied to distance-regular graphs to conclude
that these are geometric, i.e., the collinearity graph of a
partial linear space.
.LP
.B Lemma.
.I
If $GAM$ contains an $s$-claw, then
$k >= s( lam + 1) - Binom(s,2) ( mu - 1)$.
$eop$
.R
.LP
[Godsil also shows, following Metsch, that if $GAM$ contains
an $s$-claw and $lam >= (2m-1)( mu -1)$, then
$k >= s( lam + 1) - s(s-1) sup 2$.]
.LP
.B Proposition.
.I
Let $GAM$ be a distance-regular graph with smallest eigenvalue $-m$.
If $GAM$ does not have $s$-claws for $s > m$ and
$lam >= (2m-1)( mu -1)$, then $GAM$ is the collinearity graph
of a partial linear space with lines of size $1 + k/m$
and nexus $e = 1 + k/m - b sub 1 /(m-1)$.
In particular, $m$, $k/m$ and $b sub 1 /(m-1)$ are integers.
$eop$
.LP
This proposition rules out the intersection arrays
$"{"140,104,1;1,4,140"}"$ $(v = 3807)$ and
$"{"104,78,1;1,3,104"}"$ $(v = 2835)$ - the latter because
the resulting partial linear space must be a $GQ(26,4)$ minus
a spread (cf. Proposition 12.5.2), but no $GQ(26,4)$ exists.
ADDITION.
On p. 153, after Proposition 4.3.6, add: `The situation of
this proposition is also discussed in
.SC Matsumoto
.[[
Matsumoto fundamental group locally Hamming graph
.]].'
ADDITION.
On p. 153, after Corollary 4.3.7, add:
.LP
In the terminology of Chapter 11, this corollary implies
that the codes $pi sup -1 ( gam )$ are uniformly regular,
and then Theorem 11.1.6 says that if $GAM$ is distance-regular,
then these codes are completely regular. (Similar observations were
also made by
.SC Rif\*`a & Huguet
.[[
Rif\*`a Huguet 1990 classification completely regular codes
.]].)
CORRECTION.
On p. 154, the last line of Corollary 4.3.8, replace `halved' by `folded'.
ADDITION.
On p. 155 at the end of \(sc4.3B, add:
.LP
In fact,
.SC Nomura
.[[
Nomura distance regular graphs Hamming type part0
.]]
shows that if $lam != 2$ then no condition on $a sub 3$ is necessary,
i.e., if $mu = 2$, $c sub 3 = 3$, $lam != 2$ and $a sub 2 = 2 lam$,
then we have a quotient of a Hamming graph.
.SC Nomura
.[[
Nomura local structure Hamming type part2
.]]
shows that if $mu = 2$, $c sub 3 = 3$ and $d( gam , del ) = 3$, then
$C( gam , del )$ is either a cube or the direct product
$K sub 2 times K sub 1,1,2$, and the latter can occur only for $lam = 2$.
.SC Coolsaet
.[[
Coolsaet local structure
.]]
shows that if $lam = mu = 2$, $a sub 2 = 4$, then $GAM$ is locally
a union of triangles, hexagons and heptagons. Furthermore, that a
vertex together with a triangle, hexagon or heptagon in its neighbourhood
is contained in a unique $K sub 4$, Shrikhande or Klein subgraph of $GAM$,
respectively, where the Klein graph is the unique distance-regular graph
with intersection array $"{"7,4,1;1,2,7"}"$ (cf. p. 386, Remark (iv)).
He concludes that there is no distance-regular graph with intersection array
$"{"13,10,7;1,2,7"}"$.
ADDITION.
On p. 157 add a new Section 4.3E.
.Na "E" "Pappus graphs"
.B
4.3.16. Proposition
.R
.SC (Koolen
.[[
Koolen condition 1992
.]]).
.I
Let $GAM$ be a graph with $a sub 1 = a sub 2 = a sub 3 = 0$
and $c sub 2 = 1$, $c sub 3 = 2$ and $c sub 4 = 3$.
Then any pair of vertices at mutual distance 4 determines a unique
geodetically closed Pappus subgraph (with intersection array
$"{"3,2,2,1;1,1,2,3"}"$ on $18$ points).
.LP
.B Proof.
Let $d( gam , del ) = 4$. Then $C( gam , del ) minus "{" gam , del "}"$
induces a 12-gon $( alpha sub 0 alpha sub 1 ... alpha sub 11 alpha sub 0 )$
with $gam adj alpha sub 4t+1$ and $del adj alpha sub 4t+3$ ($t = 0,1,2$).
We have $d( alpha sub 0 , alpha sub 3 ) = 3$ and
$GAM ( alpha sub 3 ) ca GAM sub 2 ( alpha sub 0 ) =
"{" del , alpha sub 2 "}"$,
so $d( alpha sub 0 , alpha sub 4 ) = 4$.
We have $d( alpha sub 5 , del ) = 3$ and
$GAM ( del ) ca GAM sub 2 ( alpha sub 5 ) =
"{" alpha sub 3 , alpha sub 7 "}"$,
so $d( alpha sub 5 , alpha sub 11 ) = 4$. In the 12-gon
$C( alpha sub 5 , alpha sub 11 ) minus "{" alpha sub 5 , alpha sub 11 "}"$
we see $alpha sub 0 adj beta sub 0 adj alpha sub 6$ and
$alpha sub 4 adj beta sub 2 alpha sub 10$ for certain vertices
$beta sub 0 , beta sub 2$. Similarly we find
$alpha sub 2 adj beta sub 1 alpha sub 8$. In the 12-gon
$C( alpha sub 0 , alpha sub 4 ) minus "{" alpha sub 0 , alpha sub 4 "}"$
we see $beta sub 0 adj eps beta sub 2$, and similarly
we find $beta sub 1 adj eps prime adj beta sub 0$.
But $"{" alpha sub 0 , alpha sub 6 , eps , eps prime "}" ib
GAM ( beta sub 0 ) ca GAM sub 3 ( alpha sub 3 )$, so $eps = eps prime$
and we have found a Pappus subgraph on 18 vertices.
.Eop
.B
4.3.17. Corollary.
.I
The numbers $1o4 b sub 1 b sub 2 b sub 3$ and $smover(1,36) v k sub 4$
are integral.
.Eop
.R
This rules out the existence of distance-regular graphs with intersection
array $"{"7,6,6,5,4,3; 1,1,2,3,4,7"}"$ ($v = 686$, $k sub 4 = 210$).
.Na "F" "General"
.SC Suzuki
.[[
Suzuki strongly closed 1995
.]]
investigates strongly closed subgraphs of distance-regular graphs.
(A subgraph $DEL$ of $GAM$ is strongly closed if for $gam , del mo GAM$,
with $d( gam , del ) = i$, also $GAM ( del ) ca GAM sub { <= i } ( gam )
ib DEL$.)
ADDITION
On p. 160 add to the Remarks: Yamazaki & Nomura announce (email, 940301): `Let
$GAM$ be a bipartite distance-regular graph with valency $k >= 3$
and diameter $d >= 3$. Let $theta$ be an eigenvalue of $GAM$ and $f$
the multiplicity w.r.t. $theta$. If $k = f$, then $GAM$ is 2-homogeneous.'
and `Let $GAM$ be a bipartite 2-homogeneous distance-regular graph
with valency $k >= 3$ and diameter $d >= 3$. Then $GAM$ is a $d$-cube or
the antipodal 2-cover with $d <= 5$.'
ADDITION.
On p. 161 add before the last sentence of the Remark: `Conversely,
a completely regular clique $C$ of $GAM$ with covering radius $d-1$
satisfies (28) with equality.'
CORRECTION.
On p. 162 replace equations (30c) and (30d) by
.EQ C \fR(30c)\fP
f - i >= roman "min" ( 2 + 1o2 b sub i , b sub i )
~~ "if" ~~ lam = 1 ~ "or" ~ mu = 1,
.EN
.EQ C \fR(30d)\fP
f - i >= roman "min" ( 2 + lam over { lam + 1 } b sub i , b sub i )
~~ "if" ~~ lam > mu = 1 ~ "and" ~ i <= 1.
.EN
ADDITION.
On p. 163 add after Remark (iv):
.LP
(v)
.SC Zhu
.[[
Zhu 1989 multiplicity four
.]]
(Proposition 3.5 (i)) notices that a coconnected distance-regular
graph with an eigenvalue of multiplicity $f > 2$ also satisfies
$lam <= 1o2 (f-2)(f+1)$. (The proof is the same as that for the
inequality $b sub 1 <= 1o2 (f-2)(f+1)$ above - all one uses of the
subgraph $DEL$ in the proof above is that $d( del , gam sub l )$
is known for $del mo DEL$ and $0 <= l <= i$, and that $DEL$ has
diameter at most 2 in $GAM$.)
ADDITION.
On p. 165 add at the end of line 8: `and the Ivanov-Ivanov-Faradjev graph'.