3,317 research outputs found
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
The Cyclic Groups with them-DCI Property
AbstractFor a finite groupGand a subsetSofGwhich does not contain the identity ofG,let Cay(G,S)denote the Cayley graph ofGwith respect toS.If, for all subsetsS, TofGof sizem,Cay(G,S)≅Cay(G,T)impliesSα=Tfor someα∈Aut(G), thenGis said to have them-DCI property. In this paper, a classification is presented of the cyclic groups with them-DCI property, which is reasonably complete
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