We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the automorphism group.

The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. In 2014 Babai proved that the automorphism group of a strongly regular graph with n vertices has minimal degree $\ge cn$, with known exceptions. Strongly regular graphs correspond to distance-regular graphs of diameter 2. Babai conjectured that Hamming and Johnson graphs are the only primitive distance-regular graphs of diameter $d\ge 3$ whose automorphism group has sublinear minimal degree. We confirm this conjecture for non-geometric primitive distance-regular graphs of bounded diameter. We also show if the primitivity assumption is removed, then only one additional family of exceptions arises, the crown graphs. We settle the geometric case in a companion paper.

我们证明具有支配距离的距离正则图是频谱扩展器。证明的关键要素是交点号上的新不等式。我们使用谱带隙来研究自同构群的结构。

置换基团的最小程度ģ不是通过非同一性元件固定点的最小数量ģ。最小程度的下限对G具有强烈的结构性影响。2014年，Babai证明了具有n个顶点的强正则图的自同构群具有最小的程度$\ge Cñ$，但已知例外。强正则图对应于直径2的距离正则图。Babai推测Hamming和Johnson图是唯一的原始直径正则图$d\ge 3$其自同构群具有亚线性最小度。对于有界直径的非几何原始距离正则图，我们证实了这一猜想。我们还表明，如果除去了原始性假设，那么只会出现一个额外的例外族，即冠状图。我们将几何案例定为随行文件。