# I. N. Belousov. Shilla distance-regular graphs with $b_2 = sc_2$

A Shilla graph is a distance-regular graph $\Gamma$ of diameter 3 whose second eigenvalue is $a=a_3$. A Shilla graph has intersection array $\{ab,(a+1)(b-1),b_2;1,c_2,a(b-1)\}$. J. Koolen and J. Park showed that, for a given number $b$, there exist only finitely many Shilla graphs. They also found all possible admissible intersection arrays of Shilla graphs for $b\in \{2,3\}$. Earlier the author together with A.A. Makhnev studied Shilla graphs with $b_2=c_2$. In the present paper, Shilla graphs with $b_2=sc_2$, where $s$ is an integer greater than $1$, are studied. For Shilla graphs satisfying this condition and such that their second nonprincipal eigenvalue is $-1$, five infinite series of admissible intersection arrays are found. It is shown that, in the case of Shilla graphs without triangles in which $b_2=sc_2$ and $b<170$, only six admissible intersection arrays are possible. For a $Q$-polynomial Shilla graph with $b_2=sc_2$, admissible intersection arrays are found in the cases $b=4$ and $b=5$, and this result is used to obtain a list of admissible intersection arrays of Shilla graphs for $b\in\{4,5\}$ in the general case.

Keywords: distance-regular graph, graph automorphism.

The paper was received by the Editorial Office on June 26, 2018.

Funding Agency: This work was supported by the Russian Science Foundation (project no.~14-11-00061-П).

Ivan Nikolaevich Belousov, Cand. Sci. (Phis.-Math.), Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: i_belousov@mail.ru.

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