On automorphism groups of AT4(7, 9,r)-graphs and their local subgraphs

The paper is devoted to the problem of classification of AT4$(p,p+2,r)$-graphs. An example of an AT4$(p,p+2,r)$-graph with $p=2$ is provided by the Soicher graph with intersection array $\{56, 45, 16,1;1,8, 45, 56\}$. The question of existence of AT4$(p,p+2,r)$-graphs with $p>2$ is still open. One task in their classification is to describe such graphs of small valency. We investigate the automorphism groups of a hypothetical AT4$(7,9,r)$-graph and of its local graphs. The local graphs of each AT4$(7,9,r)$-graph are strongly regular with parameters $(711,70,5,7)$. It is unknown whether a strongly regular graph with these parameters exists. We show that the automorphism group of each AT4$(7,9,r)$-graph acts intransitively on its arcs. Moreover, we prove that the automorphism group of each strongly regular graph with parameters $(711,70,5,7)$ acts intransitively on its vertices.