文摘
This paper deals with the Cayley graph ae0e33aadd5ea8a387817d686" 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 beb5f304c548bf2159b00" title="Click to view the MathML source">2(n+1). The proof uses several properties of the subgraph 85a903f68577759a856ef36d9083" title="Click to view the MathML source">Γ of ae0e33aadd5ea8a387817d686" title="Click to view the MathML source">Cay(Symn,Tn) induced by the set Tn. In particular, 85a903f68577759a856ef36d9083" title="Click to view the MathML source">Γ is a 2(n−2)-regular graph whose automorphism group is Dn+1,85a903f68577759a856ef36d9083" title="Click to view the MathML source">Γ has as many as n+1 maximal cliques of size 2, and its subgraph 9d3d474f688e7d1354dd3986b8" 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 n+1 with regular Cayley maps on 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 Symn.