On split graphs with four distinct eigenvalues

It is a well-known fact that a graph of diameter d has at least d+1 eigenvalues. A graph is d-extremal, if it has diameter d and exactly d+1 eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have diameter at most 3. We obtain a complete classification of the connected bidegreed 3-extremal split graphs using the association of split graphs with combinatorial designs. We also construct certain families of non-bidegreed 3-extremal split graphs.