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

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