Jurišić, Aleksandar ; Koolen, Jack ; Terwilliger, Paul
We consider a distance-regular graph ▫$\Gamma$▫ with diameter ▫$d \ge 3$▫ and eigenvalues ▫$k = \theta_0 > \theta_1 >...> \theta_d$▫. We show the intersection numbers ▫$a_1,b_1$▫ satisfy ▫$$(\theta_1 ...+ \frac{k}{a_1+1}) (\theta_d +\frac{k}{a_1+1} \ge -\frac{ka_1b-1}{(a_1+1)^2}$$▫. We say ▫$\Gamma$▫ is tight whenewer ▫$\Gamma$▫ 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 ▫$a_1 \ne 0$▫, ▫$a_d=0$▫, and ▫$\gamma$▫ is 1-homogeneous in the sense of Namura. 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.
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