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. d S L -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 d ˆ , then we say it is (d ˜ , d ˆ) -bidegreed. In this paper, we present a complete classification of the connected bidegreed 3 S L -extremal split graphs using the association of split graphs with combinatorial designs. [ABSTRACT FROM AUTHOR]