A Complete Characterization of Bidegreed Split Graphs with Four Distinct alpha-Eigenvalues

It is a well-known fact that a graph of diameter d has at least d + 1 eigenvalues. A graph is d-extremal (resp. d(alpha)-extremal) if it has diameter d and exactly d + 1 distinct eigenvalues (resp. alpha-eigenvalues), and a graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have a diameter of at most three. If all vertex degrees in a split graph are either (d) over tilde or (d) over cap, then we say it is (d) over tilde or (d) over cap -bidegreed. In this paper, we present a complete classification of the connected bidegreed 3(alpha)-extremal split graphs using the association of split graphs with combinatorial designs. This result is a natural generalization of Theorem 4.6 proved by Goldberg et al. and Proposition 3.8 proved by Song et al., respectively.