We consider undirected graphs without loops and multiple edges. Previously, V. P. Burichenko and A. A. Makhnev [1] found intersection arrays of distance-regular locally cyclic graphs with the number of vertices at most 1000. It is shown that the automorphism group of a graph with intersection array {15, 12, 1; 1, 2, 15}, {35, 32, 1; 1, 2, 35}, {39, 36, 1; 1, 2, 39}, or {42, 39, 1; 1, 3, 42} (such a graph enters the above-mentioned list) acts intransitively on the set of its vertices.
Similar content being viewed by others
References
V. P. Burichenko and A. A. Makhnev “On amply regular locally cyclic graphs,” Modern Problems in Mathematics, Proc. 42nd All-Russian School–Conference of Young Scientists, Institute of Mathematics and Mechanics, UB RAS, Yekaterinburg (2011), pp. 181-183.
V. P. Burichenko and A. A. Makhnev, “On automorphisms of distance-regular graph with intersection array {15, 12, 1; 1, 2, 15},” Dokl. Ross. Akad. Nauk, 445, No. 4, 375-379 (2012).
L. Yu. Tsiovkina, “On automorphisms of a graph with intersection array {35, 32, 1; 1, 2, 35},” Sib. El. Mat. Izv., 9, 285-293 (2012).
I. N. Belousov and A. A. Makhnev, “Automorphism groups of antipodal distance-regular graphs with at most 1000 vertices,” Mal’tsev Readings (2015), p. 87.
A. A. Makhnev and M. S. Nirova, “On automorphisms of distance-regular graph with intersection array {51, 48, 8; 1, 4, 36},” Dokl. Ross. Akad. Nauk, 450, No. 1, 19-23 (2013).
L. Yu. Tsiovkina, “On automorphisms of a graph with intersection array {27, 24, 1; 1, 8, 27},” Sib. El. Mat. Izv., 10, 689-698 (2013).
A. A. Makhnev and Tsiovkina, “On automorphisms of distance-regular graph with intersection array {42, 39, 1; 1, 3, 42},” Dokl. Ross. Akad. Nauk, 441, No. 3, 305-309 (2011).
A. V. Zavarnitsine, “Finite simple groups with narrow prime spectrum,” Sib. El. Math. Rep., 6, 1-12 (2009).
The GAP Group, GAP—Groups, Algorithms, and Programming, Vers. 4.8.7 (2017); http://www.gap-system.org
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Russian Science Foundation, project No. 14-11-00061.
Translated from Algebra i Logika, Vol. 56, No. 4, pp. 395-405, July-August, 2017.
Rights and permissions
About this article
Cite this article
Belousov, I.N., Makhnev, A.A. Automorphism Groups of Small Distance-Regular Graphs. Algebra Logic 56, 261–268 (2017). https://doi.org/10.1007/s10469-017-9447-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-017-9447-4