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Automorphisms of a Distance-Regular Graph with Intersection Array {30,22,9;1,3,20}
K. S. Efimova, A. A. Makhnevbc a Ural State University of Economics, Ekaterinburg
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
c Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Abstract:
A distance-regular graph Γ of diameter 3 is called a Shilla graph if it has the second eigenvalue θ1=a3. In this case a=a3 divides k and we set b=b(Γ)=k/a. Koolen and Park obtained the list of intersection arrays for Shilla graphs with b=3. There exist graphs with intersection arrays {12,10,5;1,1,8} and {12,10,3;1,3,8}. The nonexistence of graphs with intersection arrays {12,10,2;1,2,8}, {27,20,10;1,2,18}, {42,30,12;1,6,28}, and {105,72,24;1,12,70} was proved earlier. In this paper, we study the automorphisms of a distance-regular graph Γ with intersection array {30,22,9;1,3,20}, which is a Shilla graph with b=3. Assume that a is a vertex of Γ, G=Aut(Γ) is a nonsolvable group, ˉG=G/S(G), and ˉT is the socle of ˉG. Then ˉT≅L2(7), A7, A8, or U3(5). If Γ is arc-transitive, then T is an extension of an irreducible F2U3(5)-module V by U3(5) and the dimension of V over F3 is 20, 28, 56, 104, or 288.
Keywords:
Shilla graph, graph automorphism.
Received: 02.03.2020 Revised: 26.05.2020 Accepted: 15.06.2020
Citation:
K. S. Efimov, A. A. Makhnev, “Automorphisms of a Distance-Regular Graph with Intersection Array {30,22,9;1,3,20}”, Trudy Inst. Mat. i Mekh. UrO RAN, 26, no. 3, 2020, 23–31; Proc. Steklov Inst. Math. (Suppl.), 315, suppl. 1 (2021), S89–S96
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https://www.mathnet.ru/eng/timm1742 https://www.mathnet.ru/eng/timm/v26/i3/p23
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Abstract page: | 200 | Full-text PDF : | 48 | References: | 40 | First page: | 1 |
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