V.V. Bitkina^a*, A.A. Makhnev^b**

^aNorth Ossetian State University, Vladikavkaz, 362025 Russia

^bInstitute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620990 Russia

E-mail: *bviktoriyav@mail.ru, **makhnev@imm.uran.ru

Received December 1, 2016

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Abstract

J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue ≤ t for the given positive integer t. This problem is reduced to the des­cription of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the non-principal eigenvalue t for t = 1, 2,…

In the paper “Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3”, A.A. Makhnev and D.V. Paduchikh found the arrays of intersections of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue t such as 2 < t ≤ 3. The graphs with intersection arrays {125,96,1;1,48,125}, {176,150,1;1,25,176}, and {256,204,1;1,51,256} remain unexplored.

In this paper, we have found the possible orders and the structures of subgraphs of the fixed points of automorphisms of a distance-regular graph with the intersection array {125,96,1;1,48,125}. It has been proved that the neighborhoods of the vertices of this graph are pseudogeometric graphs for GQ(4,6). Composition factors of the automorphism group of a distance-regular graph with the intersection array {125,96,1;1,48,125} have been determined.

Keywords: distance-regular graph, automorphism groups of graph

Acknowledgments. This study was supported by the Russian Science Foundation (project no. 15-11-10025) (theorem 1 and corollary), as well as by the Agreement no. 02.A03.21.0006 of August 27, 2013 between the Ministry of Education and Science of the Russian Federation and the Ural Federal University (theorem 2).

References

1. Makhnev A.A., Paduchikh D.V. Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3. Dokl. Math., 2015, vol. 92, no. 2, pp. 568–571. doi: 10.1134/S1064562415050191.

2. Brouwer A.E., Cohen A.M., Neumaier A. Distance-Regular Graphs. New York, Springer, 1989. 485 p.

3. Cameron P.J. Permutation groups. London Math. Soc. Stud. Texts. Cambridge, Cambridge Univ. Press, 1999, no. 45. 232 p.

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

5. Zavarnitsine A.V. Finite simple groups with narrow prime spectrum. Sib. Elektron. Mat. Izv., 2009, vol. 6, pp. 1–12. (In Russian)

For citation: Bitkina V.V., Makhnev A.A. On automorphisms of distance-regular graph with intersection array {125; 96; 1; 1; 48; 125}. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 1, pp. 13–20. (In Russian)

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