For a distance-regular graph Γ of diameter 3, the graph Γ_i can be strongly regular for i = 2 or 3. Finding the parameters of Γ_i from the intersection array of Γ is a direct problem, and finding the intersection array of Γ from the parameters of Γ_i is the inverse problem. The direct and inverse problems were solved earlier by A. A. Makhnev and M. S. Nirova for i = 3. In the present paper, we solve the inverse problem for i = 2: given the parameters of a strongly regular graph Γ₂, we find the intersection array of a distance-regular graph Γ of diameter 3. It is proved that Γ₂ is not a graph in the half case. We also refine Nirova’s results on distance-regular graphs Γ of diameter 3 for which Γ₂ and Γ₃ are strongly regular. New infinite series of admissible intersection arrays are found: {r² + 3r +1, r(r +1), r + 2; 1, r + 1, r(r + 2)} for odd r divisible by 3 and {2r² + 5r + 2, r(2r + 2), 2r + 3; 1, 2r + 2, r(2r + 3)} for r indivisible by 3 and not congruent to ±1 modulo 5.

中文翻译：

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距离正则图理论中的反问题

对于直径为3的距离正则图Γ，图表Γ_我可以是强正则对于我= 2或3。查找Γ的参数_我从Γ的交叉阵列的直接的问题，并发现Γ的交点阵列从Γ的参数_我是逆问题。直接和逆问题是由AA Makhnev和MS Nirova较早解决了我= 3。在本论文中，我们解决反演问题为我= 2：给出一个强正则图Γ的参数₂，我们发现交点阵列直径的距离正则图Γ的3证明了Γ ₂在一半情况下不是图。我们还完善Nirova对距离正则图Γ结果直径3条，其中Γ ₂和Γ ₃是强正则。找到新的无限系列的可容许交集：{ r ² + 3 r +1，r（r +1），r + 2; 1，r + 1，r（r + 2）}，奇数r可被3和{2 r ² + 5 r + 2，整除，r（2 r + 2），2 r + 3; 1，2 r + 2，r（2r + 3）}，其中r被3整除，且不等于±1模5。