1 research outputs found
Pentavalent symmetric graphs of order twice a prime power
A connected symmetric graph of prime valency is {\em basic} if its
automorphism group contains no nontrivial normal subgroup having more than two
orbits. Let be a prime and a positive integer. In this paper, we
investigate properties of connected pentavalent symmetric graphs of order
, and it is shown that a connected pentavalent symmetric graph of order
is basic if and only if it is either a graph of order , , ,
or a graph of three infinite families of Cayley graphs on generalized dihedral
groups -- one family has order with or , one family
has order with , and the other family has order .
Furthermore, the automorphism groups of these basic graphs are computed.
Similar works on cubic and tetravalent symmetric graphs of order have
been done.
It is shown that basic graphs of connected pentavalent symmetric graphs of
order are symmetric elementary abelian covers of the dipole \Dip_5,
and with covering techniques, uniqueness and automorphism groups of these basic
graphs are determined. Moreover, symmetric \mz_p^n-covers of the dipole
\Dip_5 are classified. As a byproduct, connected pentavalent symmetric graphs
of order are classified