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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)

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.

UDC: 519.17

Received: 01.03.2024
Revised: 05.06.2024
Accepted: 18.06.2024



© Steklov Math. Inst. of RAS, 2024