It is a well-known fact that a graph of diameter d has at least $d+1$ eigenvalues (A.E. Brouwer, W.H. Haemers (2012) [2]). A graph is d-extremal (resp. $d_{SL}$-extremal) if it has diameter d and exactly $d+1$ distinct eigenvalues (resp. signless Laplacian eigenvalues), and a graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have diameter at most three. If all vertex degrees in a split graph are either $\tilde{d}$ or $\hat{d}$, then we say it is $(\tilde{d},\hat{d})$-bidegreed. In this paper, we present a complete classification of the connected bidegreed $3_{SL}$-extremal split graphs using the association of split graphs with combinatorial designs.

众所周知，直径为d的图至少有$d+1$特征值 (AE Brouwer, WH Haemers (2012) [2])。图是d -极值（分别为$d_{秒升}$-extremal) 如果它的直径为d并且正好$d+1$不同的特征值（resp。无符号拉普拉斯特征值），如果一个图的顶点集可以划分为一个集团和一个稳定集，那么它就会被分割。这样的图最多有三个直径。如果分割图中的所有顶点度都是$\stackrel{～}{d}$ 要么 $\widehat{d}$，那么我们说它是 $(\stackrel{～}{d},\widehat{d})$-双学位。在本文中，我们提出了连通双学位的完整分类$3_{秒升}$- 使用分裂图与组合设计的关联的极值分裂图。