On split graphs with three or four distinct (normalized) Laplacian eigenvalues

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.