A non-trivial automorphism g of a graph ? is called semiregular if the only power g^i fixing a vertex is the identity mapping, and it is called quasi-semiregular if it fixes one vertex and the only power g^i fixing another vertex is the identity mapping. In this paper, we prove that K_4, the Petersen graph and the Coxeter graph are the only connected cubic arc-transitive graphs admitting a quasi-semiregular automorphism, and K_5 is the only connected tetravalent 2-arc-transitive graph admitting a quasi-semiregular automorphism. It will also be shown that every connected tetravalent G-arc-transitive graph, where G is a solvable group containing a quasi-semiregular automorphism, is a normal Cayley graph of an abelian group of odd order.
COBISS.SI-ID: 1541113028
A graph ? of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding semiregular automorphism is a cycle, have been studied, at least for the few smallest possible valences. The main theme of this paper is the question of the existence of such bicirculants for higher valences. It is proved that infinite families of edge-transitive examples of valence 6 exist and among them infinitely many arc-transitive as well as infinitely many half-arc-transitive members are identified.
COBISS.SI-ID: 1541402820
A Cayley graph of a group H is a finite simple graph ? such that Aut(?) contains a subgroup isomorphic to H acting regularly on V(?), while a Haar graph of H is a finite simple bipartite graph ? such that Aut(?) contains a subgroup isomorphic to H acting semiregularly on V(?) and the H-orbits are equal to the bipartite sets of ?. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that D6, D8, D10 in Q8 are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph.
COBISS.SI-ID: 1541356740