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 subgraphs. The local subgraphs 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.

中文翻译：

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AT4（7，9，r）-图及其局部子图的自同构群

本文致力于AT4（p，p + 2，r）图的分类问题。具有交点数组{56，45，16，1;的Soicher图提供了p = 2的AT4（p，p + 2，r）-图的示例。1、8、45、56}。p > 2的AT4（p，p + 2，r）-图的存在性问题仍然存在。它们的分类中的一项任务是描述这样的小价态图。我们研究假设的AT4（7，9，r）-图及其局部子图的自同构群。每个AT4（7、9，r）-图具有参数（711、70、5、7）的强规则。尚不知道是否存在具有这些参数的强规则图形。我们显示每个AT4（7，9，r）-图的自同构群在其弧上不传递地起作用。此外，我们证明了每个具有参数（711、70、5、7）的强正则图的自同构群在其顶点上不传递。