文摘
This paper deals with the Cayley graph e5840ae0e33aadd5ea8a387817d686" title="Click to view the MathML source">Cay(Symn,Tn), where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that Aut(Cay(Symn,Tn)) is the product of the left translation group and a dihedral group Dn+1 of order bbeb5f304c548bf2159b00" title="Click to view the MathML source">2(n+1). The proof uses several properties of the subgraph a856ef36d9083" title="Click to view the MathML source">Γ of e5840ae0e33aadd5ea8a387817d686" title="Click to view the MathML source">Cay(Symn,Tn) induced by the set 8d202de035579722fc" title="Click to view the MathML source">Tn. In particular, a856ef36d9083" title="Click to view the MathML source">Γ is a baf007a8e83cb296c87" title="Click to view the MathML source">2(n−2)-regular graph whose automorphism group is Dn+1,a856ef36d9083" title="Click to view the MathML source">Γ has as many as bb9338ff15" title="Click to view the MathML source">n+1 maximal cliques of size 2, and its subgraph 8e7d1354dd3986b8" title="Click to view the MathML source">Γ(V) whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of Dn+1 of order bb9338ff15" title="Click to view the MathML source">n+1 with regular Cayley maps on ba66" title="Click to view the MathML source">Symn is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non-t-balanced regular Cayley map on ba66" title="Click to view the MathML source">Symn.