Large vertex-transitive and Cayley graphs with given degree and diameter

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 kmk. By comparison with other available results in this area we show that, for sufficiently large d and k such that k⩽d−2, our family gives the current largest known Cayley graphs of degree d and diameter k.