Geodesic
On $Q$-polynomial Shilla graphs with $b = 4$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 176-186

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Shilla graphs introduced by J. H. Koolen and J. Park are considered. In the problem of finding feasible intersection arrays of Shilla graphs with a fixed parameter $b$, $Q$-polynomial graphs play an important role. For such graphs, the smallest eigenvalue is the minimum possible for the third nonprincipal eigenvalue. Intersection arrays of $Q$-polynomial graphs were found for $b=3$ in 2010 by Koolen and Park and for $b\in\{4,5\}$ in 2018 by Belousov. In particular, it is known that a $Q$-polynomial Shilla graph with $b=4$ has intersection array $\{104,81,27;1,9,78\}$, $\{156,120,36;1,12,117\}$, or $\{20(q-2),3(5q-9),2q;1,2q,15(q-2)\}$, where $q=6,9,18$. We prove that distance-regular graphs with intersection arrays $\{80,63,12;1,12,60\}$, $\{140,108,18;1,18,105\}$, and $\{320,243,36;1,36,240\}$ do not exist.
Keywords: Shilla graph, distance-regular graphs
Mots-clés : $Q$-polynomial graph. 
A. A. Makhnev; I. N. Belousov; M. P. Golubyatnikov. On $Q$-polynomial Shilla graphs with $b = 4$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 176-186. http://geodesic.mathdoc.fr/item/TIMM_2022_28_2_a13/
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