The Uniqueness of the Triangular Association Scheme

Strongly regular graphs, partial geometries and partially balanced designs

“…Substituting these values in (1.10) we get n > 8. Thus for these special values of r, k, t our result is equivalent to the uniqueness of the triangular scheme for n > 8, a result first proved by Connor [9]. In fact our result may be interpreted as a generalized uniqueness theorem as explained in § 12.…”

“…// in a pseudo-geometric graph with characteristics (r,k,t) 9 there exists a set Σ of cliques K lf K 2 , , K b , satisfying axioms A*l and A*2, and if k > r, then G is geometrisable (r, k, t), the vertices of G, and the cliques of Σ being the points and lines of the corresponding geometry.…”

“…In particular the partial geometry (r, k 9 1) becomes a net of degree r and order k when t = r -1. A pseudogeometric graph with characteristics (r, k, r -1) may be defined to be a pseudo-net graph, of degree r and order k. Bruck [5] has proved a series of lemmas for pseudo-net graphs and in particular has shown that a pseudo-net graph of degree r and order k is geometrisable (r, k, r -1) if (1.9) k > i (r -l)(r 3 -r 2 + r + 2) .…”

On the tetrahedral graph

“…For simplicity let us denote the vertices of G by the natural numbers. (1,3,2) in C' n and {2, , n -2} on the lattice points {{i, 2,1) | 4 <Ξ i ^ n} in this order, {w -1, , 2n -5} on {(i, 3,1) | 4 ^ i ^ w}, and {2n -4, , Sn -8} on {(i, 3, 2) I 4 ^ i ^ w}.…”

“…Conner [2], Shrikhande [7], Hoffman [3,4] and Li-chien [5,6] showed that (A0)-(A3) completely characterize linegraphs of complete graphs except for n = 8 where 3 nonisomorphic graphs satisfying (AO)-(A3) exist. Bose and Laskar [1] took up the similar problem concerning unordered triplets chosen from n symbols we mentioned above.…”