ENUMERATION INTERSECTION ARRAYS OF SHILLA GRAPHS WITH B = 6

Alexander A. Makhnev     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)
Ivan N. Belousov     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)
Mikhail P. Golubyatnikov     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108; Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russian Federation)

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 arrays

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References


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DOI: http://dx.doi.org/10.15826/umj.2025.2.012

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