TIGHT DISTANCE - REGULAR GRAPHSAleksandar

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 (a 1 + 1) 2 : 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 6 = 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 innnite families and nine sporadic examples of tight distance-regular graphs are given.