A graph is split if its vertex set can be partitioned into a clique and an independent set. A split graph is -bidegreed if each of its vertex degrees is equal to either or . Each connected split graph is of diameter at most 3. In 2017, Nikiforov proposed the -matrix, which is the convex combination of the adjacency matrix and the diagonal matrix of vertex degrees of the graph . It is well-known that a connected graph of diameter contains at least distinct -eigenvalues. A graph is said to be -extremal with respect to its -matrix if the graph is of diameter having exactly distinct -eigenvalues. In this paper, using the association of split graphs with combinatorial designs, the connected -extremal (resp. -extremal) bidegreed split graphs are classified. Furthermore, all connected bidegreed split graphs of diameter 2 having just 4 distinct -eigenvalues are identified.
"COMPLETE CHARACTERIZATION OF THE BIDEGREED SPLIT GRAPHS WITH THREE OR FOUR DISTINCT -EIGENVALUES." Rocky Mountain J. Math. 53 (5) 1571 - 1585, October 2023. https://doi.org/10.1216/rmj.2023.53.1571