M.P. Golubyatnikov. Automorphisms of a distance-regular graph with intersection array {35, 32, 28; 1, 4, 8} ... P. 54-63

We continue the study of automorphisms of distance-regular locally cyclic graphs with at most 4096 vertices (the intersection arrays of such graphs were found earlier by A.A. Makhnev and M.S. Nirova). Let $\Gamma$ be a distance-regular graph with intersection array $\{35,32,28;1,4,8\}$. Then it has eigenvalue $\theta_2=-1$ and the graph $\bar \Gamma_3$ is pseudogeometric for the net $pG_8(35,8)$ and has parameters $(1296,315,90,72)$. We study possible automorphisms of such graphs. In particular, for a graph $\Gamma$ with intersection array $\{35,32,28;1,4,8\}$ and $G={\rm Aut}(\Gamma)$, it is proved that $\pi(G)\subseteq \{2,3,5,7\}$. Further, if a nonsolvable group $G={\rm Aut}(\Gamma)$ acts transitively on the vertex set of a graph with intersection array $\{35,32,28;1,4,8\}$ and $\bar T$ is the socle of the group $\bar G=G/S(G)$, then $G=S(G)G_a$, $\bar T_a\cong A_5$, and $\bar T_{a,b}\cong A_4$ for some vertices $a\in \Gamma$ and $b\in [a]$.

Keywords: strongly regular graph, distance-regular graph, graph automorphism.

The paper was received by the Editorial Office on Februery 27, 2018.

Mikhail Petrovich Golubyatnikov, undergraduate student, Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: mike_ru1@mail.ru.

REFERENCES

1.   Makhnev A.A., Nirova M.S. On distance-regular graphs with λ = 2. J. Siberian Federal Univ., 2014, vol. 7, no. 2, pp. 204–210.

2.   Makhnev A.A., Paduchikh D.V. On automorphisms of a distance-regular graph with intersection array {24,21,3;1,3,18}. Algebra i Logika, 2012, vol. 51, no. 4, pp. 476–495.

3.   Makhnev A.A., Nirova M.S. On automorphisms of a distance-regular graph with intersection array {51,48,8;1,4,36}. Dokl. Math. 2013, vol. 87, no. 3, pp. 269–273. doi: 10.7868/S0869565213130045 .

4.   Makhnev A.A., Paduchikh D.V. Automorphisms of a distance-regular graph with intersection array {18,15,9;1,1,10}. Commun. Math. Stat., 2015, vol. 3, no. 4, pp. 527–534.
doi: 10.1007/s40304-015-0072-z .

5.   J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson. Atlas of finite groups. Oxford: Clarendon Press, 1985, 252 p.

6.   Brouwer A.E., Cohen A.M., Neumaier A. Distance-regular graphs. Berlin; Heidelberg; N Y: Springer-Verlag, 1989, 495 p. ISBN: 0387506195 .

7.   Cameron P.J. Permutation groups. Ser. London Math Soc. Student Texts, vol. 45, Cambridge: Cambridge Univ. Press, 1999, 232 p. ISBN: 0-521-65302-9 .

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

9.   Behbahani M., Lam C. Strongly regular graphs with nontrivial automorphisms. Discrete Math., 2011, vol. 311, no. 2-3, pp. 132–144. doi: 10.1016/j.disc.2010.10.005 .

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

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