Distance-Regular Graph with Iintersection Array {140,108,18;1,18,105} Does not Exist

Makhnev, A. A. , Nirova, M. S.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 2.

Abstract:
Distance-regular graph \(\Gamma\) of diameter 3 having the second eigenvalue \(\theta_1=a_3\) is called Shilla graph. In this case \(a=a_3\) devides \(k\) and we set \(b=b(\Gamma)=k/a\). Jurishich and Vidali found intersection arrays of \(Q\)-polynomial Shilla graphs with \(b_2=c_2\): \(\{2rt(2r+1),(2r-1)(2rt+t+1),r(r+t);1,r(r+t),t(4r^2-1)\}\). 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 \(b_2=c_2\) (\(t=2r^2-1\)): \(\{2r(2r^2-1)(2r+1),(2r-1)(2r(2r^2-1)+2r^2),r(2r^2+r-1);1,r(2r^2+r-1),(2r^2-1)(4r^2-1)\}\). 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.

Keywords: distance-regular graph, triangle-free graph, triple intersection numbers

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