Tight Distance-Regular Graphs

We consider a distance-regular graph with diameter d 3 and eigenvalues k = ₀ > ₁ > ... > _d . We show the intersection numbers a ₁, b ₁ satisfy We say is tight whenever is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show is tight if and only if the intersection numbers are given by certain rational expressions involving d independent parameters. We show is tight if and only if a ₁ 0, a _d = 0, and is 1-homogeneous in the sense of Nomura. We show is tight if and only if each local graph is connected strongly-regular, with nontrivial eigenvalues –1 – b ₁(1 + ₁)^–1 and –1 – b ₁(1 + _d )^–1. Three infinite families and nine sporadic examples of tight distance-regular graphs are given.