Alladi Ramakrishnan Hall

Automorphism group of Cayley graphs generated by transpositions

Ashwin Ganesan

Vidyalankar Institute of Technology, Mumbai

I shall present some recent results on the automorphism group
of some families of Cayley graphs. Let H be a group and let S be a
generating set for H. The automorphism group of every Cayley graph
Cay(H,S) contains the following two subgroups: the right regular
representation R(H) and the set Aut(H,S) of automorphisms of H that
fixes S setwise. A Cayley graph Cay(H,S) is said to be normal if it has
the smallest possible (full) automorphism group in the sense that it has
no other automorphisms besides R(H) Aut(H,S). An open problem in the
literature is to determine, given H and S, the normality and
automorphism group of Cay(H,S). We obtain some sufficient conditions
for a Cayley graph of the symmetric group generated by transpositions to
be normal. The automorphism group of the star graphs, bubble-sort
graphs and modified bubble-sort graphs are special cases of our result.
We also investigate the edge-transitivity and isomorphism classes of
Cayley graphs generated by transpositions.