Enumeration of intersection arrays of Shilla graphs with $b=6$

Alexander A. Makhnev^a, Ivan N. Belousov^a, Mikhail P. Golubyatnikov<sup>ab

a</sup> N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
^b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg

Abstract: Let $\Gamma$ be a distance-regular graph of diameter $3$, and let $\theta_1$ be its second eigenvalue. The graph $\Gamma$ is called a Shilla graph if $\theta_1=a_3$. In this case, $\theta_1={(a_1+\sqrt{a_1^2+4k})}/{2}$, and $a=a_3$ divides $k$. We set $b=b(\Gamma)=k/a$. J. H. Koolen and J. Park found the intersection arrays of Shilla graphs with $b\le 3$. J. Cai, I. N. Belousov, and A. A. Makhnev enumerated the intersection arrays of Shilla graphs with $b=4$. H. Li, I. N. Belousov, and A. A. Makhnev found the intersection arrays of Shilla graphs with $b=5$. In this paper, we enumerate the intersection arrays of Shilla graphs with $b=6$.

Keywords: Distance-regular graph, Shilla graph, Intersection array.

Language: English

DOI: 10.15826/umj.2025.2.012

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