It is well known to us that a graph of diameter $l$ has at least $l+1$ eigenvalues. A graph is said to be Laplacian (resp, normalized Laplacian) $l$‐extremal if it is of diameter $l$ having exactly $l+1$ distinct Laplacian (resp, normalized Laplacian) eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Each split graph is of diameter at most 3. In this paper, we completely classify the connected bidegreed Laplacian (resp, normalized Laplacian) 2‐extremal (resp, 3‐extremal) split graphs using the association of split graphs with combinatorial designs. Furthermore, we identify connected bidegreed split graphs of diameter 2 having just four Laplacian (resp, normalized Laplacian) eigenvalues.

中文翻译：

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在具有三个或四个不同（规范化）拉普拉斯特征值的分裂图上

我们都知道直径图 $升$ 至少有 $升+\mathrm{1个}$特征值。图被称为拉普拉斯算子（resp，归一化拉普拉斯算子）$升$-极值如果是直径$升$ 完全有 $升+\mathrm{1个}$不同的拉普拉斯算子（resp，归一化的拉普拉斯算子）特征值。如果图的顶点集可以划分为集团和稳定集，则该图将被拆分。每个拆分图的直径最大为3。在本文中，我们使用拆分图与组合设计的关联关系，对连接的双级Laplacian（resp，归一化Laplacian）2-极端（resp，3-extremal）拆分图进行了完全分类。此外，我们确定了仅具有四个Laplacian（resp，归一化Laplacian）特征值的直径2的连通双度分割图。