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

Let vt(d; 2) be the largest order of a vertex-transitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d; 2) b d+2 2 cd d+2 2 e. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows that vt(d; 2) 8 9 (d + 1 2 ) 2 for all d of the form d = (3q \Gamma 1)=2 where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d = 7 we obtain as a special case the Hoffman-Singleton graph, and for d = 11 and d = 13 we have new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively. 1 Introduction The well-known degree/diameter problem asks for determining the largest possible number n(d; k) of vertices..

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

Let vt(d; 2) be the largest order of a vertex-transitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d; 2) b d+2 2 cd d+2 2 e. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows that vt(d; 2) 8 9 (d + 1 2 ) 2 for all d of the form d = (3q \Gamma 1)=2 where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d = 7 we obtain as a special case the Hoffman-Singleton graph, and for d = 11 and d = 13 we have new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively. 1 Introduction The well-known degree/diameter problem asks for determining the largest possible number n(d; k) of vertic..