Restrictions on classical distance-regular graphs

Abstract

Let \(\varGamma \) be a distance-regular graph with diameter \(d \ge 2\). 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 $$\begin{aligned} b_i= & {} ([d]_b - [i]_b)(\beta - \alpha [i]_b) \qquad \text {and} \qquad c_{i+1} = [i+1]_b (1 + \alpha [i]_b)\\&\quad (0 \le i \le d-1), \end{aligned}$$ where \([i]_b := 1 + b + \cdots + b^{i-1}\). Apart from the well-known families, there are many sets of classical parameters for which the existence of a corresponding graph is still open. It turns out that in most such cases we have either \(\alpha = b-1\) or \(\alpha = b\). For these two cases, we derive bounds on the parameter \(\beta \), which give us complete classifications when \(b = -2\). Distance-regular graphs with classical parameters are antipodal iff \(b=1\) and \(\beta =1+\alpha [d-1]_b\). If we drop the condition \(b=1\), it turns out that one obtains either bipartite or tight graphs. For the latter graphs, we find closed formulas for the parameters of the CAB partitions and the distance partition corresponding to an edge. Finally, we find a two-parameter family of feasible intersection arrays for tight distance-regular graphs with classical parameters \((d,b,b-1,b^{d-1})\) (primitive iff \(b \ne 1\)) and apply our results to show that it is realized only by d-cubes (\(b = 1\)).