I.N. Belousov, A.A. Makhnev, and M.S. Nirova described $Q$-polynomial distance-regular graphs $\Gamma$ of diameter 3 for which the graphs $\Gamma_2$ and $\Gamma_3$ are strongly regular. Set $a=a_3$. A graph $\Gamma$ has type (I) if $c_2+1$ divides~$a$, type (II) if $c_2+1$ divides $a+1$, and type (III) if $c_2+1$ divides neither $a$ nor $a+1$. If $\Gamma$ is a graph of type (II), then $a+1=w(c_2+1)$, $t^2=w(w(c_2+1)+c_2)$, and either
(i) $w=s^2$, $t^2=s^2(s^2(c_2+1)+c_2)$, $(s^2(c_2+1)+c_2$ is the square of an integer $u$, $c_2=(u^2-s^2)/(s^2+1)$, $t=su$, and $a=(u^2s^2-1)/(s^2+1)$ or
(ii) $c_2=sw$, $t^2=w^2(sw+1+s)$, $sw+1+s$ is the square of an integer $u$, $c_2=(u^2-1)w/(w+1)$, $t=uw$, $a=(u^2w^2-1)/(w+1)$, and $\Gamma$ has intersection array
$$\left\{ \frac{u^3w^2+u^2w^2+uw-1}{w+1},\frac{(u^2-1)uw^2}{w+1},\frac{(u^2w+1)w}{w+1};1,\frac{(u^2-1)w}{w+1},\frac{(u^2w+1)uw}{w+1}\right\}.$$ If a graph of type (IIii) is such that $w=u$, then it has intersection array $\{w^4+w-1,w^4-w^3,(w^2-w+1)w;$ $1,w(w-1),(w^2-w+1)w^2\}$. We prove that graphs with such intersection arrays do not exist for even $w$.
Keywords: distance-regular graph, $Q$-polynomial graph
Received September 10, 2019
Revised November 7, 2019
Accepted November 11, 2019
Funding Agency: This work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).
Aleksandr Alekseevich Makhnev, Dr. Phys.-Math. Sci., 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
Mikhail Petrovich Golubyatnikov, doctoral student, 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: mike_ru1@mail.ru
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Cite this article as: A.A.Makhnev, M.P.Golubyatnikov. Nonexistence of certain Q-polynomial distance-regular graphs, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 136–141.
