Description
Let \Gamma be a distance-regular graph of diameter d. It is said to have classical parameters (d, b, \alpha, \beta) when its intersection array \{b_0,b_1,\dots,b_{d-1};c_1,c_2,\dots,c_d\} satisfies b_i = ([d] - [i])(\beta - \alpha [i]) and c_{i+1} = [i+1] (1 + \alpha [i]) (0 \le 1 \le d-1), where [j] := 1+b+\cdots+b^{j-1} is the b-analogue of j.
One can say that a distance-regular graph \Gamma with d \ge 2 and eigenvalues k=\theta_0>\theta_1>\cdots >\theta_d 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}. We study tight graphs with classical parameters. The known examples include the Johnson graph J(2d, d) (b=1, \alpha=1, \beta=d), the halved 2d-cube {1 \over 2} H(2d, 2) (b=1, \alpha=2, \beta=2d-1) and the Gosset graph (d=3, b=1, \alpha=4, \beta=9). We show that a distance-regular graph of diameter d \ge 3 with classical parameters is tight if and only if \beta = 1+\alpha [d-1] and b, \alpha > 0. For such graphs, we explicitly compute the parameters of the CAB partitions and the equitable partition corresponding to an edge. We also make a conjecture regarding local graphs of classical tight distance-regular graphs.
Then we find a two parameter family of feasible intersection arrays for d \ge 4, \alpha=b-1 > 0 and \beta = b^{d-1}. They correspond to primitive tight distance-regular graphs -- note that the Patterson graph (related to the Suzuki group) is the only known such graph. The smallest example of the above family is obtained for d=4, b=2 and has intersection array \{120,98,60,8;1,6,28,120\}, which is listed in the extended tables of feasible intersection arrays by Brouwer. Finally, we prove that there exists no distance-regular graph with intersection array belonging to this family.
Joint work with Aleksandar Jurišić.
Primary author
Dr
Janoš Vidali
(University of Primorska and University of Ljubljana)
