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.