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For any d ≥ 5 and k ≥ 3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d − 3)/3) k. By comparison with other available results in this area we show that, for all sufficiently large d and k, our family gives the current largest known Cayley graphs of degree d and diameter k.

Let) 2 , (d C ,) 2 , (d AC , and) 2 , (d CC be the largest order of a Cayley graph of a group, an Abelian group, and a cyclic group, respectively, of diameter 2 and degree d. The currently known best lower bounds on these parameters are 2 /) 1 () 2 , (2 d d C for degrees 1 2 q d where q is an odd prime power,) 4)(8 / 3 () 2 , (2 d d AC where 2 4… (More)

Consider a composition of two regular coverings π 1 : Γ 0 → Γ 1 and π 2 : Γ 1 → Γ 2 of graphs, given by voltage assignments α 1 , α 2 on Γ 1 , Γ 2 in groups G 1 and G 2 , respectively. In the case when π 2 • π 1 is regular we present an explicit voltage assignment description of the composition in terms of G 1 , G 2 , α 1 , α 2 , and walks in Γ 1 .

We consider random Cayley digraphs of order n with uniformly distributed generating set of size k. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as n → ∞ and k = f (n). We find a sharp phase transition from 0 to 1 as the order of growth of f (n) increases past √ n log n. In particular, if f (n)… (More)

We present an upper bound on the number of vertices in graphs of given degree and diameter 3 that arise as lifts of dipoles with voltage assignments in Abelian groups. Further, we construct a family of Cayley graphs of degree d = 3m − 1 and diameter k ≥ 3 of order km k. By comparison with other available results in this area we show that, for sufficiently… (More)