Abstract
A graph
$\Gamma $
is called
$(G, s)$
-arc-transitive if
$G \le \text{Aut} (\Gamma )$
is transitive on the set of vertices of
$\Gamma $
and the set of s-arcs of
$\Gamma $
, where for an integer
$s \ge 1$
an s-arc of
$\Gamma $
is a sequence of
$s+1$
vertices
$(v_0,v_1,\ldots ,v_s)$
of
$\Gamma $
such that
$v_{i-1}$
and
$v_i$
are adjacent for
$1 \le i \le s$
and
$v_{i-1}\ne v_{i+1}$
for
$1 \le i \le s-1$
. A graph
$\Gamma $
is called 2-transitive if it is
$(\text{Aut} (\Gamma ), 2)$
-arc-transitive but not
$(\text{Aut} (\Gamma ), 3)$
-arc-transitive. A Cayley graph
$\Gamma $
of a group G is called normal if G is normal in
$\text{Aut} (\Gamma )$
and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if
$\Gamma $
is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either
$\Gamma $
is normal or G is one of the groups
$\text{PSL}_2(11)$
,
$\text{M} _{11}$
,
$\text{M} _{23}$
and
$A_{11}$
. However, it was unknown whether
$\Gamma $
is normal when G is one of these four groups. We answer this question by proving that among these four groups only
$\text{M} _{11}$
produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.
Cubic Non-Cayley Vertex-Transitive Bi-Cayley Graphs over a Regular $p$-Group