A.A. Makhnev, D.V. Paduchikh. On distance-regular graphs with intersection arrays $\{q^2-1,q(q-2),q+2;1,q,(q+1)(q-2)\}$ ... P. 146-156

If a distance-regular graph $\Gamma$ of diameter 3 contains a maximal locally regular 1-code that is last subconstituent perfect, then $\Gamma$ has intersection array $\{a(p+1),cp,a+1;1,c,ap\}$ or $\{a(p+1),(a+1)p,c;1,c,ap\}$, where $a=a_3$, $c=c_2$, and $p=p^3_{33}$ (Jurisic, Vidali). In the first case, $\Gamma$ has eigenvalue $\theta_2=-1$ and the graph $\Gamma_3$ is pseudogeometric for $GQ(p+1,a)$. If $a=c+1$, then the graph $\bar\Gamma_2$ is pseudogeometric for $pG_2(p+1,2a)$. If in this case the pseudogeometric graph for the generalized quadrangle $GQ(p+1,a)$ has quasi-classical parameters, then $\Gamma$ has intersection array $\{q^2-1,q(q-2),q+2;1,q,(q+1)(q-2)\}$ (Makhnev, Nirova). In this paper, we find possible automorphisms of a graph with intersection array $\{q^2-1,q(q-2),q+2;1,q,(q+1)(q-2)\}$.

Keywords: distance-regular graph, generalized quadrangle, graph automorphism

Received September 9, 2020

Revised December 20, 2020

Accepted January 11, 2021

Funding Agency: This work was supported by the Russian Foundation for Basic Research — the National Natural Science Foundation of China (project no. 20-51-53013).

Aleksandr Alekseevich Makhnev, Dr. Phys.-Math. Sci., RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: makhnev@imm.uran.ru

Dmitrii Viktorovich Paduchikh, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: dpaduchikh@gmail.com

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Cite this article as: A.A. Makhnev, D.V. Paduchikh. On distance-regular graphs with intersection arrays $\{q^2-1,q(q-2),q+2;1,q,(q+1)(q-2)\}$, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 1, pp. 146–156.