Title:
Vertex-primitive
digraphs
having
vertices
with
almost
equal
neighbourhoods
Speaker: | Gabriel Verret |
Affiliation: | University of Auckland |
Room: | MC 6486 |
Abstract:
A
permutation
group
G
on
V
is
transitive
if
for
every
x,
y
in
V
there
exists
g
in
G
mapping
x
to
y.
The
group
G
is
called
primitive
if,
in
addition,
it
preserves
no
nontrivial
partition
of
V.
Let
X
be
a
vertex-primitive
digraph,
that
is,
its
automorphism
group
acts
primitively
on
its
vertex-set.
It
is
not
hard
to
see
that,
in
this
case,
X
cannot
have
two
distinct
vertices
with
equal
neighbourhoods,
unless
X
is
in
some
sense
trivial.
I
will
discuss
some
recent
results
about
the
case
when
X
has
two
vertices
with
“almost”
equal
neighbourhoods,
and
how
these
results
were
used
to
answer
a
question
of
Araújo
and
Cameron
about
synchronising
groups.
(This
is
joint
work
with
Pablo
Spiga.)