A Shilla graph is a distance-regular graph Γ of diameter 3 whose second eigenvalue is a = a₃. A Shilla graph has intersection array {ab, (a + 1)(b − 1), b₂; 1, c₂, a(b − 1)}. J. Koolen and J. Park showed that, for a given number b, there exist only finitely many Shilla graphs. They also found all possible admissible intersection arrays of Shilla graphs for b ∈ {2, 3}. Earlier the author together with A. A. Makhnev studied Shilla graphs with b₂ = c₂. In the present paper, Shilla graphs with b₂ = sc₂, where s is an integer greater than 1, are studied. For Shilla graphs satisfying this condition and such that their second nonprincipal eigenvalue is −1, five infinite series of admissible intersection arrays are found. It is shown that, in the case of Shilla graphs without triangles in which b₂ = sc₂ and b < 170, only six admissible intersection arrays are possible. For a Q-polynomial Shilla graph with b₂ = sc₂, admissible intersection arrays are found in the cases b = 4 and 5, and this result is used to obtain a list of admissible intersection arrays of Shilla graphs for b ∈ {4, 5} in the general case.

中文翻译：

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b 2 = sc 2的新罗距离正则图

希拉（Shilla）图是直径3的距离正则图Γ，其第二特征值是a = a ₃。Shilla图的交集为{ ab，（a + 1）（b − 1），b ₂ ; 1，c ₂，a（b -1）}。J. Koolen和J. Park表明，对于给定的数b，仅存在有限的许多新罗图。他们还发现新罗图表为所有可能的可容许相交阵列b ∈{2,3}。之前，作者与AA Makhnev一起研究了b ₂ = c _2的新罗图。在本文中，研究了b ₂ = sc _2的Shilla图，其中s是大于1的整数。对于满足此条件并使其第二个非本征特征值为-1的Shilla图，找到了五个无限的容许交点阵列序列。结果表明，在没有三角形的Shilla图的情况下，其中b ₂ = sc ₂且b <170，只有六个可允许的交点阵列是可能的。对于b ₂ = sc ₂的Q多项式Shilla图，在这种情况下找到了可允许的交集b = 4和5，和该结果被用于获得新罗图表用于受理相交阵列的列表b ∈{4,5}中的一般情况。