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 b226b0aceaf3efad29425b0f12b12a4" title="Click to view the MathML source">Dn+1 of order 2(n+1). The proof uses several properties of the subgraph 83" 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, 83" 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 83d6cbf1a3c0b630d7ee47" title="Click to view the MathML source">Dn+1,83" title="Click to view the MathML source">Γ has as many as a104fbb9338ff15" title="Click to view the MathML source">n+1 maximal cliques of size 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 b226b0aceaf3efad29425b0f12b12a4" title="Click to view the MathML source">Dn+1 of order a104fbb9338ff15" title="Click to view the MathML source">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.