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 af3efad29425b0f12b12a4" title="Click to view the MathML source">Dn+1 of order 2(n+1). The proof uses several properties of the subgraph Γ of e5840ae0e33aadd5ea8a387817d686" title="Click to view the MathML source">Cay(Symn,Tn) induced by the set Tn. In particular, Γ is a baf007a8e83cb296c87" title="Click to view the MathML source">2(n−2)-regular graph whose automorphism group is Dn+1,Γ has as many as 8ff15" title="Click to view the MathML source">n+1 maximal cliques of size 8fd4ceb9812fb0820fe0b6b795" title="Click to view the MathML source">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 af3efad29425b0f12b12a4" title="Click to view the MathML source">Dn+1 of order 8ff15" 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-a01354d9c4c520140af09b" title="Click to view the MathML source">t-balanced regular Cayley map on ba66" title="Click to view the MathML source">Symn.