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Distance-regular graph with intersection array {140,108,18;1,18,105} does not exist
A. A. Makhneva, M. S. Nirovab a N. N. Krasovskii Institute of Mathematics and Mechanics, 16 S. Kovalevskaja St., Ekaterinburg 620990, Russia
b Kabardino-Balkarian State University, 173 Chernyshevsky St., Nalchik 360004, Russia
Abstract:
Distance-regular graph Γ of diameter 3 having the second eigenvalue θ1=a3 is called Shilla graph. In this case a=a3 devides k and we set b=b(Γ)=k/a. Jurishich and Vidali found intersection arrays of Q-polynomial Shilla graphs with b2=c2: {2rt(2r+1),(2r−1)(2rt+t+1),r(r+t);1,r(r+t),t(4r2−1)}. But many arrays in this series are not feasible. Belousov I. N. and Makhnev A. A. found a new infinite series feasible arrays of Q-polynomial Shilla graphs with b2=c2 (t=2r2−1): {2r(2r2−1)(2r+1),(2r−1)(2r(2r2−1)+2r2),r(2r2+r−1);1,r(2r2+r−1),(2r2−1)(4r2−1)}. If r=2 then we have intersection array {140,108,18;1,18,105}. In the paper it is proved that graph with this intersection array does not exist.
Key words:
distance-regular graph, triangle-free graph, triple intersection numbers.
Received: 14.12.2020
Citation:
A. A. Makhnev, M. S. Nirova, “Distance-regular graph with intersection array {140,108,18;1,18,105} does not exist”, Vladikavkaz. Mat. Zh., 23:2 (2021), 65–69
Linking options:
https://www.mathnet.ru/eng/vmj764 https://www.mathnet.ru/eng/vmj/v23/i2/p65
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Abstract page: | 106 | Full-text PDF : | 22 | References: | 34 |
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