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 2(n+1). The proof uses several properties of the subgraph 85a903f68577759a856ef36d9083" title="Click to view the MathML source">Γ of e5840ae0e33aadd5ea8a387817d686" 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 e83cb296c87" title="Click to view the MathML source">2(n−2)-regular graph whose automorphism group is b630d7ee47" title="Click to view the MathML source">Dn+1,85a903f68577759a856ef36d9083" title="Click to view the MathML source">Γ has as many as n+1 maximal cliques of size b69aa38fd4ceb9812fb0820fe0b6b795" title="Click to view the MathML source">2, and its subgraph 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-a01354d9c4c520140af09b" title="Click to view the MathML source">t-balanced regular Cayley map on Symn.