We consider the hyperspace(K,d) of all non-empty compact subsets of X endowed with the Hausdorff metric d,we will show that for any compact totally disconnected subset Z,there exists some K in K such that Z is homeomorphic to K and K is a transitive point.

In this paper,we prove that a connected cubic vertex-transitive graph of order 2p~2(p a prime)is either a Cayley graph or isomorphic to the generalized Petersen graph p(p~2,t), where t~2≡-1(mod p~2

In fact, researching the symmetry property of the Coset graph is more significant than the Cayleygraph, because every vertex-transitive graph is always some Coset graph of its full automorphism group.

In fact, researching the symmetry of Sabidussi coset graphs has more general significance than doing so for the Cayley graph because every vertex-transitive graph is always some coset graph of its full automorphism group.

Firstly,we obtain some equivalent characterization for Cayley graphs of completely simple semigroup which is vertex-transitive.

Secondly,we give a sufficient condition for the cyclic optimality of vertex- or edgetransitive graphs,that is:any connectedκ-regular vertex-transitive graph with k≥4 and girth at least five is cyclically optimal;any connected edge-transitive graph with minimum degree at least four and order at least six is cyclically optimal.

If Sz(q)≤G≤Aut(Sz(q)),where q=22n+1,then G cannot act point-transitively on S.

It is proved that if Soc(G)=Ree(q),where q=32n+1 and n>0,then G cannot act point-transitively on a finite projective plane.