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Proceedings of the Institute of Mathematics of the NAS of Belarus, 2024, Volume 32, Number 1, Pages 57–63
(Mi timb383)
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DISCRETE MATHEMATICS AND MATHEMATICAL CYBERNETICS
On distance regular graphs with diameter $3$ and degree $44$
M. Chena, A. A. Makhnevbc, V. S. Kliminc a Universitet provintsii Khaynan’, Kheykou, China
b N. N. Krasovsky Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russia
c Ural Federal University, Yekaterinburg, Russia
Abstract:
Distance-regular graph $\Gamma$ with strongly regular graphs $\Gamma_2$ and $\Gamma_3$ has intersection array $\{r(c_2+1)+a_3$, $rc_2$, $a_3+1$; 1, $c_2$, $r(c_2+1)\}$ (M. S. Nirova). For distance-regular graph with diameter 3 and degree 44 there are 7 fisiable intersection arrays. For each of them the graph $\Gamma_3$ is strongly regular. For intersection array $\{44, 30, 5; 1, 3, 40\}$ we have $a_3=4$, $c_2=3$ and $r = 10$, $\Gamma_2$ has parameters $(540, 440, 358, 360)$ and $\Gamma_3$ has parameters $(540, 55, 10, 5)$. This graph does not exist (Koolen-Park). For intersection array $\{44, 35, 3; 1, 5, 42\}$ the graph $\Gamma_3$ has parameters $(375, 22, 5, 1)$. Graph $\Gamma_3$ does nor exist (local subgraph is the union of isolated $6$-cliques). In this paper it is proved that distance-regular graphs with intersection arrays $\{44, 36, 5; 1, 9, 40\}$, $\{44, 36, 12; 1, 3, 33\}$ and $\{44, 42, 5; 1, 7, 40\}$ do not exist.
Keywords:
distance-regular graph, strongly regular graph, triple intersection numbers.
Received: 01.03.2024 Revised: 05.06.2024 Accepted: 18.06.2024
Citation:
M. Chen, A. A. Makhnev, V. S. Klimin, “On distance regular graphs with diameter $3$ and degree $44$”, Proceedings of the Institute of Mathematics of the NAS of Belarus, 32:1 (2024), 57–63
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https://www.mathnet.ru/eng/timb383 https://www.mathnet.ru/eng/timb/v32/i1/p57
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