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Abstract:We examine the existing constructions of the smallest known vertex-transitive graphs of a given degree and girth 6. It turns out that most of these graphs can be described in terms of regular lifts of suitable quotient graphs. A further outcome of our analysis is a precise identification of which of these graphs are Cayley. We also investigate higher level of transitivity of the smallest known vertex-transitive graphs of a given degree and girth 6 and relate their constructions to near-difference sets. ᭧

a b s t r a c tWe derive an upper bound on the number of vertices in regular graphs of given degree and diameter arising as regular coverings of dipoles over abelian groups.