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.

REFERENCES

1.   Brouwer A.E., Cohen A.M., Neumaier A. Distance-regular graphs. Berlin; Heidelberg; N Y: Springer-Verlag, 1989, 495 p. ISBN: 0387506195 .

2.   Koolen J.H., Park J. Shilla distance-regular graphs. Europ. J. Comb., 2010, vol. 31, no. 8, pp. 2064–2073. doi: 10.1016/j.ejc.2010.05.012 .

3.   Makhnev A.A., Nirova M.S. Distance-regular Shilla graphs with $b_2 = c_2$. Mat. Zametki, 2018, vol. 103, no. 5, pp. 730–744 (in Russian). doi: 10.4213/mzm11503 .

4.   Belousov I.N., Makhnev A.A. To the theory of Shilla graphs with $b_2 = c_2$. Sib. Elektron. Mat. Izv., 2017, vol. 14, pp. 1135–1146. doi: 10.17377/semi.2017.14.097 .

5.   Coolsaet K. Distance-regular graph with intersection array {21,16,8;1,4,14} does not exist. Europ. J. Comb., 2005, vol. 26, no. 5, pp. 709–716. doi: 10.1016/j.ejc.2004.04.005 .