Jurišić, Aleksandar ; Koolen, Jack ; Terwilliger, Paul
We prove the following bound for a ▫$k$▫-regular graph on ▫$n$▫ vertices with nontrivial eigenvalues from the interval ▫$[r,s]$▫ ▫$$n(k+rs) \le (k-r)(k.s).$$▫ Equality holds if and only if the graph ...is strongly regular with eigenvalues in ▫$\{k,s.r\}$▫. Nonbipartite distance-regilar graphs with diameter ▫$d \ge 3$▫ and eigenvalues ▫$k=\theta_0 > \theta_1 >...> \theta_d$▫, whose local graphs satisfy the above bound with equality for ▫$s = -1 - b_1/(\theta_1+1)$▫, and ▫$r = -1-b_1/(\theta_d+1)$▫ are called tight graphs and are characterized in many ways in our paper: Tight-regular graphs, Prepr.ser. - Univ. Ljubl. Inst. Math., 36 (1998), 622. For example, a distance-regular graph is tight if and only if it is 1-homogeneous and ▫$a_d=0$▫. We study tight graphs of small diameter. It turns out that in the case of diameter three these are precisely the Taylor graphs and in the case of antipodal diameter four these are precisely the graphs for which the Krein parameter ▫$q_{11}^4$▫ vanishes. We derive nonexistence conditions, which rule out twenty otherwise feasible arrays of distance-regular graphs fom the list in work: A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distace-regular graphs, Springer Verlag, Berlin, Heidelberg, 1989, pp. 421-425. We prove that in an antipodal distance regular graph ▫$\Gamma$▫ with diameter four and vanishing Krein parameters ▫$q_{11}^4$▫ and ▫$q_{44}^4$▫ every second subconstituent graph is again an antipodal distance-regular graph of diameter four. Finally, if ▫$\Gamma$▫ is also a double-cover, i.e., ▫$Q$▫-polynomial, then it is 2-homogeneous.
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