On \(Q\)-Polynomial Shilla Graphs with \(b=6\)

Makhnev, A. A. , Zhigang Van
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 2.

Abstract:
Distance-regular raph \(\Gamma\) of diameter 3, having the second eigenvalue \(\theta_1= a_3\) is called Shilla graph. For such graph \(a=a_3\) devides \(k\) and we set \(b = b(\Gamma) = k/a\). Further \(a_1 = a - b\) and \(\Gamma\) has intersection array \(\{ab,(a + 1)(b - 1), b_2; 1, c_2, a(b - 1)\}\). I. N. Belousov and A. A. Makhnev found feasible arrays of \(Q\)-polynomial Shilla graphs with \(b=6\): \(\{42t,5(7t+1),3(t+3);1,3(t+3),35t\}\), where \(t\in \{7,12,17,27,57\}\), \(\{312,265,48;1,24,260\}\), \(\{372,315,75;1,15,310\}\), \(\{624,525,80;1,40,520\}\), \(\{744,625,125;1,25,620\}\), \(\{930,780,150;1,30,775\}\), \(\{1794,1500,200;1,100,1495\}\) or \(\{5694, 4750,600;1,300,4745\}\). It is proved in the paper that graphs with intersection arrays \(\{372,315,75;1,15,310\}\), \(\{744,625,125;1,25,620\}\) and \(\{1794,1500,200;1,100,1495\}\) do not exist.

Keywords: distance-regular graph, Shilla graph, triple intersection numbers.

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