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The minimal degree of a permutation group $G$ is the minimum number of pointsnot fixed by non-identity elements of $G$. Lower bounds on the minimal degreehave strong structural consequences on $G$. In 2014 Babai proved that theautomorphism group of a strongly regular graph with $n$ vertices has minimaldegree $\geq c n$, with known exceptions. Strongly regular graphs correspond toprimitive coherent configurations of rank 3. We extend Babai's result toprimitive coherent configurations of rank 4. We also show that the resultextends to non-geometric distance-regular graphs of bounded diameter. Theproofs combine structural and spectral methods.

On the automorphism groups of distance-regular graphs and rank-4 primitive coherent configurations