A. A. Makhnev, D. V. Paduchikh, and M. M. Khamgokova. Automorphisms of strongly regular graphs with parameters (1305,440,115,165) ... P.232-242

A graph $\varGamma$ is called $t$-isoregular if, for any $i\le t$ and any $i$-vertex subset $S$, the number $\lvert {\varGamma(S)}\rvert$ depends only on the isomorphism class of the subgraph induced by $S$. A graph $\varGamma$ on $v$ vertices is called absolutely isoregular if it is $(v-1)$-isoregular. It is known that each $5$-isoregular graph is absolutely isoregular, and such graphs have been fully described. Each exactly $4$-isoregular graph is either a pseudogeometric graph for pG$_r(2r,2r^3+3r^2-1)$ or its complement. By Izo($r$) we denote a pseudogeometric graph for pG$_r(2r,2r^3+3r^2-1)$. Graphs Izo($r$) do not exist for a infinite set of values of $r$ ($r=3,4,6,10,\ldots$). The existence of Izo($5$) is unknown. In this work we find possible automorphisms for the neighborhood of an edge from Izo($5$).

Keywords: isoregular graph, strongly regular graph, pseudogeometric graph.

The paper was received by the Editorial Office on April 24, 2017.

A.A.Makhnev, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: makhnev@imm.uran.ru

D.V.Paduchikh, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: dpaduchikh@gmail.com

M.M.Hamgokova, Kand. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: hamgokova.madina@yandex.ru

REFERENCES

1.   Cameron P., Van Lint J. Designs, graphs, codes and their links. Cambridge: Cambridge University Press, 1981, 240 p. ISBN: 0521423856 .

2.   Bannai E., Munemasa A., Venkov B. The nonexistence of certain tight spherical designs. St. Petersburg Math. J., 2005, vol. 16, no. 4, pp. 609–625. doi: 10.1090/S1061-0022-05-00868-X .

3.   Nebe G., Venkov B. On tight spherical designs. St. Petersburg Math. J., 2013, vol. 24, no. 3, pp. 485–491. doi: 10.1090/S1061-0022-2013-01249-0 .

4.   Makhnev A.A. On nonexistence of strongly regular graphs with parameters (486,165,36,66). Ukrainian Mathematical Journal, 2002, vol. 54, no. 7, pp. 1137–1146. doi: 10.1023/A:1022066425998 .

5.   Makhnev A.A., Khamgokova M.M. Automorphisms of strongly regular graph with parameters (532,156,30,52). Sib. Elektron. Mat. Izv., 2015, vol. 12, pp. 930–939. doi: 10.17377/semi.2015.12.078 .

6.    Brouwer A.E., Haemers W.H. The Gewirtz graph: an exercize in the theory of graph spectra. European J. Combin., 1993, vol. 14, no. 5, pp. 397–407. doi: 10.1006/eujc.1993.1044 .

7.    Cameron P.J. Permutation groups. Cambridge: Cambridge University Press, 1999, 220 p. doi: 10.1017/CBO9780511623677 .

8.   Gavrilyuk A.L., Makhnev A.A., On automorphisms of distance-regular graph with the intersection array {56,45,1;1,9,56}. Dokl. Math., 2010, vol. 81, no. 3, pp. 439–442. doi: 10.1134/S1064562410030282 .

9.   MacKay M., Siran J. Search for properties of the missing Moore graph. Linear Algebra Appl., 2010, vol. 432, no. 9, pp. 2381–2398. doi: 10.1016/j.laa.2009.07.018 .

10.   Zavarnitsine A.V. Finite simple groups with narrow prime spectrum. Sib. Elektron. Mat. Izv., 2009, vol. 6, pp. 1–12.