Geodesic
On distance regular graphs with diameter $3$ and degree $44$
Trudy Instituta matematiki, Tome 32 (2024) no. 1, pp. 57-63

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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. 
M. Chen; A. A. Makhnev; V. S. Klimin. On distance regular graphs with diameter $3$ and degree $44$. Trudy Instituta matematiki, Tome 32 (2024) no. 1, pp. 57-63. http://geodesic.mathdoc.fr/item/TIMB_2024_32_1_a6/
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