9 research outputs found
Vertex-primitive groups and graphs of order twice the product of two distinct odd primes
A non-Cayley number is an integer n for which there exists a vertex-transitive graph on n vertices which is not a Cayley graph. In this paper, we complete the determination of the non-Cayley numbers of the form 2pq, where p, q are distinct odd primes. Earlier work of Miller and the second author had dealt with all such numbers corresponding to vertex-transitive graphs admitting an imprimitive subgroup of automorphisms. This paper deals with the primitive case. First the primitive permutation groups of degree 2pq are classified. This depends on the finite simple group classification. Then each of these groups G is examined to determine whether there are any non-Cayley graphs which admit G as a vertex-primitive subgroup of automorphisms, and admit no imprimitive subgroups. The outcome is that 2pq is a non-Cayley number, where
Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite -arc-transitive graph which is not a Cayley graph
A \emph{mixed dihedral group} is a group with two disjoint subgroups
and , each elementary abelian of order , such that is generated by
, and . In this paper we give a sufficient
condition such that the automorphism group of the Cayley graph \Cay(H,(X\cup
Y)\setminus\{1\}) is equal to , where is the setwise
stabiliser in \Aut(H) of . We use this criterion to resolve a
questions of Li, Ma and Pan from 2009, by constructing a -arc transitive
normal cover of order of the complete bipartite graph \K_{16,16} and
prove that it is \emph{not} a Cayley graph.Comment: arXiv admin note: text overlap with arXiv:2303.00305,
arXiv:2211.1680
A class of non-Cayley vertex-transitive graphs associated with PSL(2,p)
AbstractA construction for a class of non-Cayley vertex-transitive graphs associated with PSL(2,p) acting by right multiplication on the right cosets of a dihedral subgroup Dp−1 is presented and a description of these graphs is given