26 research outputs found
Generalized quaternion groups with the -DCI property
A Cayley digraph Cay(G,S) of a finite group with respect to a subset
of is said to be a CI-digraph if for every Cayley digraph Cay(G,T)
isomorphic to Cay(G,S), there exists an automorphism of such that
. A finite group is said to have the -DCI property for some
positive integer if all -valent Cayley digraphs of are CI-digraphs,
and is said to be a DCI-group if has the -DCI property for all . Let be a generalized quaternion group of order
with an integer , and let have the -DCI
property for some . It is shown in this paper that is
odd, and is not divisible by for any prime . Furthermore,
if is a power of a prime , then has the -DCI
property if and only if is odd, and either or .Comment: 1
Highly arc transitive digraphs
Unendliche, hochgradig bogentransitive Digraphen werden definiert und anhand von Beispielen vorgestellt. Die Erreichbarkeitsrelation und EigenschaftâZ werden definiert und unter Verwendung von Knotengraden, Wachstum und anderen Eigenschaften, die von der Untersuchung von Nachkommen von Doppelstrahlen oder Automorphismengruppen herrĂŒhren, auf hochgradig bogentransitiven Digraphen untersucht. Seifters Theoreme ĂŒber hochgradig bogentransitive Digraphen mit mehr als einem Ende, seine daherrĂŒhrende Vermutung und deren sie widerlegende Gegenbeispiele werden vorgestellt. Eine Bedingung, unter der Câhomogene Digraphen hochgradig bogentransitiv sind, wird angegeben und die Verbindung zwischen hochgradig bogentransitiven Digraphen und total unzusammenhĂ€ngenden, topologischen Gruppen wird erwĂ€hnt. Einige Bemerkungen ĂŒber die Vermutung von CameronâPraegerâWormald werden gemacht und eine verfeinerte Version vermutet. Die Eigenschaften der bekannten hochgradig bogentransitiven Digraphen werden gesammelt. Es wird festgestellt, dass einige, aber nicht alle unter
ihnen CayleyâGraphen sind. SchlieĂlich werden offen gebliebene Fragestellungen und Vermutungen zusammengefasst und neue hinzugefĂŒgt. FĂŒr die vorgestellten Lemmata, Propositionen und Theoreme sind entweder Beweise enthalten, oder Referenzen zu Beweisen werden angegeben.Infinite, highly arc transitive digraphs are defined and examples are given. The ReachabilityâRelation and Property-Z are defined and investigated on infinite, highly arc transitive digraphs using the valencies, spread and other properties arising from the investigation of the descendants of lines or the automorphism groups. Seifters theorems about highly arc transitive digraphs with more than one end, his conjecture on them and the counterexamples that disproved his conjecture, are given. A condition for Câhomogeneous digraphs to be highly arc transitve is stated and the connection between highly arc transitive digraphs and totally disconnected, topological groups is mentioned. Some notes on the CameronâPraegerâWormaldâConjecture are made and a refined conjecture is stated. The properties of the known highly arc transitive digraphs are collected, some but not all of them are Cayleyâgraphs. Finally open questions and conjectures are stated and new ones are added. For the given lemmas, propositions and theorems either proofs or references to proofs are included
On the Isomorphisms of Cayley Graphs of Abelian Groups
AbstractLet G be a finite group, S a subset of G\{1}, and let Cay (G,S) denote the Cayley digraph of G with respect to S. If, for any subset T of G\(1), Cay(G,S)â
Cay(G,T) implies that Sα=T for some αâAut(G), then S is called a CI-subset. The group G is called a CIM-group if for any minimal generating subset S of G,SâȘSâ1 is a CI-subset. In this paper, CIM-abelian groups are characterized
Finite -connected homogeneous graphs
A finite graph \G is said to be {\em -connected homogeneous}
if every isomorphism between any two isomorphic (connected) subgraphs of order
at most extends to an automorphism of the graph, where is a
group of automorphisms of the graph. In 1985, Cameron and Macpherson determined
all finite -homogeneous graphs. In this paper, we develop a method for
characterising -connected homogeneous graphs. It is shown that for a
finite -connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is
--transitive or G_v^{\G(v)} is of rank and \G has girth , and
that the class of finite -connected homogeneous graphs is closed under
taking normal quotients. This leads us to study graphs where is
quasiprimitive on . We determine the possible quasiprimitive types for
in this case and give new constructions of examples for some possible types