Uncountable families of vertex-transitive graphs of finite degree

Recently the following question was relayed [1] to the second author: What is the cardinality of the set of vertex transitive graphs of finite degree? Our aim in this short note is to show that there are 20 such graphs. Our proof is constructive and is based on ideas of B. Neumann [3]. In order to construct a large such set it is natural to turn to Cayley graphs of finitely generated groups (see e.g [2] for definitions and general facts about Cayley graphs). In 1937 B. Neumann [3] proved that the set of finitely generated groups has cardinality 20 . However, since Cayley graphs of non-isomorphic groups can be isomorphic, this alone does not prove that there are the same number of non-isomorphic Cayley graphs. We will give two uncountable families of groups and generators for which one can prove that the corresponding Cayley graphs are non-isomorphic. Our first family of graphs is based on a variation of the construction of Neumann, and its members are 4-regular. Our second family consists of cubic graphs, for which the isomorphism problem requires a bit more work. From these examples uncountable families of transitive regular graphs of any degree d ≥ 3 can be constructed as suitable products of our examples with complete, or complete bipartite, graphs.