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Cayley graphs of given degree and diameter for cyclic, Abelian, and metacyclic groups. (English) Zbl 1232.05091
Summary: Let $$CC(d,2)$$ and $$AC(d,2)$$ be the largest order of a Cayley graph of a cyclic and an Abelian group, respectively, of diameter 2 and a given degree $$d$$. There is an obvious upper bound of the form $$CC(d,2)\leq AC(d,2)\leq d^{2}/2+d+1$$. We prove a number of lower bounds on both quantities for certain infinite sequences of degrees $$d$$ related to primes and prime powers, the best being $$CC(d,2)\geq (9/25)(d+3)(d - 2)$$ and $$AC(d,2)\geq (3/8)(d^{2} - 4)$$. We also offer a result for Cayley graphs of metacyclic groups for general degree and diameter.

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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