On Automorphisms of a Graph with an Intersection Array \(\{44,30,9;1,5,36\}\)

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/x0578-3097-1488-l On Automorphisms of a Graph with an Intersection Array \(\{44,30,9;1,5,36\}\) Isakova, M. M. , Makhnev, A. A. , Chen Mingzhu Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 3. Abstract: For the set \(X\) automorphisms of the graph \(\Gamma\) let \({\rm Fix}(X)\) be a set of all vertices of \(\Gamma\) fixed by any automorphism from \(X\). There are 7 feasible intersection arrays of distance regular graphs with diameter 3 and degree 44. Early it was proved that for fifth of them graphs do not exist. In this paper it is founded possible automorphisms of distance regular graph with intersection array \(\{44,30,9;1,5,36\}\). The proof of the theorem is based on Higman’s method of working with automorphisms of a distance regular graph. The consequence of the main result is is the following: Let \(\Gamma\) be a distance regular graph with intersection array \(\{44,30,9;1,5,36\}\) and the group \(G={\rm Aut}(\Gamma)\) acts vertex-transitively; then \(G\) acts intransitively on the set arcs of \(\Gamma\). Keywords: strongly regular graph, fixed point subgraph, distance regular graph, automorphism Language: Russian Download the full text For citation: Isakova, M. M., Makhnev, A. A. and Chen Mingzhu. On Automorphisms of a Graph with an Intersection Array \(\{44,30,9;1,5,36\}\), Vladikavkaz Math. J., 2024, vol. 26, no. 3, pp. 47-55 (in Russian). DOI 10.46698/x0578-3097-1488-l + References 1. Brouwer, A. E., Cohen, A. M. and Neumaier, A. Distance-Regular Graphs, Berlin, Heidelberg, New York, Springer-Verlag, 1989. 2. Chen Mingzhu, Makhnev, A. A. and Klimov, V. S. On Distance Regular Graphs of Diameter 3 and Degrees 44, Molodezhnaya konferenciya IMM UrO RAN. Tezisy dokladov [Youth Conference of the Institute of Mechanics and Mathematics, Ural Branch of the Russian Academy of Sciences. Abstracts of Reports], Ekaterinburg, 2024, pp. 1132-1134 (in Russian). 3. Gavrilyuk, A. L. and Makhnev, A. A. On Automorphisms of a Distance-Regular Graph with Intersection Array \(\{56,45,1;1,9,56\}\), Doklady Rossiyskoy akademii nauk [Reports of the Russian Academy of Sciences], 2010, vol. 432, no. 5, pp. 512-515 (in Russian). 4. Cameron P. J. Permutation Groups, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, 1999. DOI: 10.1017/CBO9780511623677. 5. Cameron, P. J. and van Lint, J. Permutation Groups, London Mathematical Society Student Texts, vol. 22, Cambridge University Press, 1991. ← Contents of issue


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