CORRECTION.
On p. 392 the Buekenhout diagram in the middle was drawn incorrectly:
the double stroke should have been right instead of left.
.so Bdiag/A7
ADDITION.
On p. 392 add to construction D: `(This construction is discussed
in great detail by
.SC Calderbank & Wales
.[[
Calderbank Wales global code invariant 1982
.]].)'
CORRECTION.
On p. 396, line $-9$, replace `$(v,k, lam , mu )$' by `$(v,k, lam )$'.
ADDITION.
On p. 400, bottom, add: `Let $SIGMA sub n sup eps$ be
the graph on the hyperbolic points of $PG(n-1,3)$ provided with
a nondegenerate symmetric bilinear form with discriminant $eps$,
adjacent when perpendicular.
.SC Pasechnik
.[[
Pasechnik 1993 locally 3-transposition
.]]
shows that $SIGMA sub n+1 sup eps$ is the unique connected locally
$SIGMA sub n sup eps$ graph for $eps = +1$, $n >= 6$ and for
$eps = -1$, $n >= 7$. The example in this section shows that there are
at least two connected locally $SIGME sub 6 sup "\(mi"$ graphs (and Pasechnik
announces that there are precisely two).'
ADDITION.
On p. 412, Remark (iii), add: `For a characterization as locally polar graph, see
.SC Weiss & Yoshiara
.[[
Weiss Yoshiara 1990 geometric
.]].'
ADDITION.
On p. 412, add Remark (iv): `A. Juri\*Vsi\*'c, J.H. Koolen & P. Terwilliger
proved that any graph with the parameters of the Patterson graph is
locally strongly regular [pers.comm.]. Indeed, we have
$(1 + b sup \(mi )(1 + b sup \(pl ) + lam <= ( lam + 1 + b sup \(mi )( lam
+ 1 + b sup \(pl ) / k$ with equality only if the local graph is strongly
regular (with eigenvalues $- 1 - b sup \(mi$ and $- 1 - b sup \(pl$).
[Proof: By Theorem 4.4.3 we have
$sum ( eta + 1 + b sup \(mi )( eta + 1 + b sup \(pl ) <= 0$
where the sum is over the eigenvalues of the local graph distinct from
the valency. On the other hand, $sum 1 = k$, $sum eta = 0$,
$sum eta sup 2 = k lam$, where this time the sum is over all
eigenvalues of the local graph.]
ADDITION.
.br
.B
13.8 A triple cover of the strongly regular Suzuki graph
.LP
.SC Soicher
.[[
Soicher 1993 three distance-regular
.]]
discovered the existence of a distance-transitive graph
with automorphism group $3 . roman "Suz" :2$
and point stabilizer $G sub 2 (4) :2$ and diagram
.KS
[diagram]
.KE
In this case the $r$-covers of the underlying strongly regular graph
(with parameters $(v,k, lam , mu ) = (1782,416,100,96)$)
are feasible for $r mo "{"2,3,4,6,8,12,16,24,32"}"$.
However, the $mu$-graphs in this strongly regular graph
are edge-regular with $(v,k, lam ) = (96,20,12)$, and one easily
sees that their components must have size at least 28.
Since the above-mentioned 3-cover exists, it follows that
these $mu$-graphs all have three components, each of size 32.
Thus, this graph has no other distance-regular covers.
[And in fact, the $mu$-graphs are $3(2. 1o2 2 sup 5 )$: three copies
of the 2-coclique extension of the halved 5-cube.]