A Note on Large Graphs of Diameter Two and Given Maximum Degree,

@article{McKay1998ANO,
  title={A Note on Large Graphs of Diameter Two and Given Maximum Degree,},
  author={Brendan D. McKay and Mirka Miller and Jozef Sir{\'a}n},
  journal={J. Comb. Theory, Ser. B},
  year={1998},
  volume={74},
  pages={110-118}
}
Letvt(d,2) be the largest order of a vertex-transitive graph of degreedand diameter 2. It is known thatvt(d,2)=d2+1 ford=1,2,3, and 7; for the remaining values ofdwe havevt(d,2)?d2?1. The only knowngenerallower bound onvt(d,2), valid forall d, seems to bevt(d,2)??(d+2)/2? ?(d+2)/2?. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows thatvt(d,2)?(8/9)(d+12)2for alldof the formd=(3q?1)/2, whereqis a prime power congruent with 1 (mod 4). The construction… 
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