Two families of graphs that are Cayley on nonisomorphic groups

A number of authors have studied the question when a graph can be represented as Cayley on more than one nonisomorphic group. The work to date has focussed few special situations: groups are $p$-groups; order $pq$; graphs normal; or both abelian. In this paper, we construct two infinite families graphs, each which is an abelian group and nonabelian These include smallest examples such that had not appeared in other results.