A complete characterization of bidegreed split graphs with four distinct signless Laplacian eigenvalues

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. dSL-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 d˜ or dˆ, then we say it is (d˜,dˆ)-bidegreed. In this paper, we present a complete classification of the connected bidegreed 3SL-extremal split graphs using the association of split graphs with combinatorial designs.