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

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


1.   Gavrilyuk A.L., Makhnev A.A., Paduchikh D.V. On distance-regular graphs in which neighborhoods of vertices are strongly regular. Dokl. Math., 2013, vol. 88, no. 2, pp. 532–536. doi: 10.1134/S1064562413050116 .

2.   Brouwer A.E. Parameters of strongly regular graphs [site] . Available on http://www.win.tue.nl/~ aeb/graphs/srg/srgtab.html .

3.   Brouwer A.E., Cohen A.M., Neumaier A. Distance-regular graphs, Berlin etc: Springer-Verlag, 1989, 495 p. doi: 10.1007/978-3-642-74341-2 .

4.   Cameron P.J. Permutation groups, Cambridge, Cambridge Univ. Press, 1999, 220 p. ISBN-10: 0521653789 .

5.   Behbahani M., Lam C. Strongly regular graphs with nontrivial automorphisms. Discrete Math., 2011, vol. 311, pp. 132–144. doi: 10.1016/j.disc.2010.10.005 .

6.   Zavarnitsine A.V. Finite simple groups with narrow prime spectrum. Siberian Electr. Math. Reports, 2009, vol. 6, pp. 1–12.

7.   Guralnick R., Kunyavskii B., Plotkin E., Shalev A. Thompson-like characterizations of the solvable radical. J. Algebra, 2006, vol. 300, pp. 363–375. doi: 10.1016/j.jalgebra.2006.03.001 .