Abstract
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the groups have order $pq$; when the Cayley graphs are normal; or when the groups are both abelian. In this paper, we construct two infinite families of graphs, each of which is Cayley on an abelian group and a nonabelian group. These families include the smallest examples of such graphs that had not appeared in other results.
Highlights
A Cayley graph Cay(G, S) on a group G with connection set S, is the graph whose vertices are the elements of G, with two vertices g1 and g2 adjacent if and only if g2 = sg1 for some s ∈ S
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group
The work to date has focussed on a few special situations: when the groups are p-groups; when the groups have order pq; when the Cayley graphs are normal; or when the groups are both abelian
Summary
Abstract: A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. It is well known (first observed by Sabidussi) that a (di)graph can be represented as a Cayley (di)graph on the group G if and only if its automorphism group contains a subgroup isomorphic to G
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