Title: On edge-transitive distance-regular antipodal covers of complete graphs
| Speaker: | Ludmila Tsiovkina |
| Affiliation: | Krasovsky Institute of Mathematics and Mechanics |
| Location: | Contact Sabrina Lato for zoom link |
Abstract: A
distance-regular
antipodal
cover
of
a
complete
graph
is
equivalently
defined
as
a
connected
graph
whose
vertex
set
can
be
partitioned
into
a
set
of
$n$
blocks
(or
fibres)
of
the
same
size
$r\ge
2$
such
that
each
block
induces
an
$r$-coclique,
the
union
of
any
two
distinct
blocks
induces
a
perfect
matching,
and
any
two
non-adjacent
vertices
lying
in
distinct
blocks
have
exactly
$\mu\ge
1$
common
neighbours;
such
a
graph
will
be
briefly
referred
to
as
an
$(n,r,\mu)$-cover.
Let
$\Gamma$
be
an
edge-transitive
$(n,r,\mu)$-cover
and
$G=Aut(\Gamma)$.
It
can
be
easily
seen
that
$G$
induces
a
$2$-homogeneous
permutation
group
$G^{\Sigma}$
on
the
set
$\Sigma$
of
fibres
of
$\Gamma$,
which
by
combination
of
two
well-known
theorems
of
Kantor
and
Burnside
is
either
almost
simple
or
affine.
In
the
talk,
I
will
overview
some
recent
results
on
classification
of
edge-transitive
$(n,r,\mu)$-covers
both
in
almost
simple
and
affine
cases.
In
particular,
I
will
show
that
non-bipartite
edge-transitive
$(n,r,\mu)$-covers
in
almost
simple
case
can
be
characterized
(in
essentially
a
uniform
way)
as
basic
graphs
of
certain
schurian association
schemes
of
quasi-simple
groups.