The paper is devoted to the problem of classification of ${\rm AT4}(p,p+2,r)$-graphs. An example of an ${\rm AT4}(p,p+2,r)$-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 ${\rm AT4}(p,p+2,r)$-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 ${\rm AT4}(7,9,r)$-graph and of its local graphs. The local graphs of each ${\rm AT4}(7,9,r)$-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 ${\rm AT4}(7,9,r)$-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.
Keywords: antipodal tight graph, strongly regular graph, automorphism
The paper was received by the Editorial Office on June 4, 2018.
Funding Agency: This work was supported by the Russian Science Foundation (project no.~14-11-00061-П).
Lyudmila Yur’evna Tsiovkina, Cand. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: l.tsiovkina@gmail.com
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