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This article is cited in 4 scientific papers (total in 4 papers)
Discrete mathematics and mathematical cybernetics
Distance-regular graph with intersection array $\{105,72,24;1,12,70\}$ does not exist
I. N. Belousov, A. A. Makhnev N.N. Krasovskii Institute of Mathematics and Mechanics of UB RAS,
16, S.Kovalevskaya str.,
Yekaterinburg, 620990, Russia
Abstract:
Distance-regular graph $\Gamma$ of diameter 3 is called Shilla graph if $\Gamma$ containes the second eigenvalue $\theta_1=a_3$. In this case $a=a_3$ devides $k$ and we set $b=b(\Gamma)=k/a$. Koolen and Park obtained the list of intersection arrays for Shilla graphs with $b=3$. A. Brouwer with coauthors proved that graph with intersection array $\{27,20,10;1,2,18\}$ does not exist. $Q$-polinomial Shilla graph with $b=3$ has intersection array $\{42,30,12;1,6,28\}$ or $\{105,72,24;1,12,70\}$. Early authors proved that graph with intersection array $\{42,30,12;1,6,28\}$ does not exist.
We prove that graph with intersection array $\{105,72,24;1,12,70\}$ does not exist.
Keywords:
distance-regular graph, Shilla graph, triple intersection numbers.
Received December 18, 2018, published February 8, 2019
Citation:
I. N. Belousov, A. A. Makhnev, “Distance-regular graph with intersection array $\{105,72,24;1,12,70\}$ does not exist”, Sib. Èlektron. Mat. Izv., 16 (2019), 206–216
Linking options:
https://www.mathnet.ru/eng/semr1050 https://www.mathnet.ru/eng/semr/v16/p206
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