Tight distance-regular graphs

We consider a distance-regular graph Gamma with diameter d greater than or equal to 3 and eigenvalues k = theta (0) > theta (1) > ... > theta (d). We show the intersection numbers a(1), b(1) satisfy (theta (1) + k/a(1) + 1) (theta (d) + k/a(1)+1) greater than or equal to - ka(1)b(1)/(a(1) + 1)(2). We say Gamma is tight whenever Gamma is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show Gamma is tight if and only if the intersection numbers are given by certain rational expressions involving d independent parameters. We show Gamma is tight if and only if a(1) not equal 0, a(d) = 0, and Gamma is 1-homogeneous in the sense of Nomura. We show Gamma is tight if and only if each local graph is connected strongly-regular, with nontrivial eigenvalues -1 - b(1)(1 + theta (1))(-1) and -1 - b(1)(1 + theta (d))(-1). Three infinite families and nine sporadic examples of tight distance-regular graphs are given.