On complete split graphs that are Laplacian $\{-1,0,1\}$-diagonalizable with bandwidth 2

We investigate complete split graphs whose Laplacian matrices are $\{-1,0,1\}$-diagonalizable with bandwidth $2$, namely, weakly Hadamard diagonalizable (WHD) graphs. We first characterize the complete split graphs that are $\{-1,0,1\}$-diagonalizable in terms of their threshold structure and Laplacian eigenbases. Next, using recursive join decompositions together with known properties of WHD graphs, we obtain sufficient conditions ensuring that a complete split graph is WHD. Finally, we provide a computational classification of all $\{-1,0,1\}$-diagonalizable complete split graphs on at most $20$ vertices, including their bandwidths and the cases that are WHD.