Characterization of some of the biggest distance-regular graphs, which do not belong to any of the infinite families (like Johnson and Hamming graphs), by using their intersection numbers. The biggest among them is the Patterson graph, which is a primitive distance-transitive and therefore also distance-regular graph on 22.880 vertices with valency 280 that we can construct from the sporadic simple finite Suzuki group. In this way we achive the characterization of this remarkable group with only seven parameters.
COBISS.SI-ID: 14632537
Triangle-free primitive distance-regular graphs G with diameter 3 and an eigenvalue t with multiplicity equal to their valency are studied. We show that it is formally self-dual (and hence Q-polynomial and 1-homogeneous). Let x,y be adjacent vertices, and z \in G_2(x) \cap G_2(y). Then the intersection number \tau_2:=|G(z)\cap G_3(x)\cap G_3(y)| is independent of x, y and z. We classify all the graphs with b_2=\tau_2 (example: the coset graph of the doubly truncated binary Golay code) and rule out an infinite family of otherwise feasible intersection arrays.
COBISS.SI-ID: 14519897