The prime spectrum of an automorphism group of an $\mathrm {AT4}(p,p+2,r)$-graph

The present paper is devoted classification of $\mathrm {AT4}(p,p+2,r)$-graphs. There a unique {AT4}(p,p+2,r)$-graph with $p=2$, namely, the distance-transitive Soicher graph intersection array $\{56, 45, 16, 1;1, 8, 56\}$, whose local graphs are isomorphic to Gewirtz graph. existence an ${p>2}$ remains open question. It known that each strongly regular parameters $\big ((p+2)(p^2+4p+2),p(p+3),p-2,p\big )$. In this paper, upper bound found for prime spectrum automorphism group such parameters, and also some restrictions obtained structure in case when $2<p$ power. As corollary, it shown there no arc-transitive {AT}(p,p+2,r)$-graphs $p\in \{11,17,27\}$.