COMPLETE CHARACTERIZATION OF THE BIDEGREED SPLIT GRAPHS WITH THREE OR FOUR DISTINCT Aα-EIGENVALUES

Abstract

A graph is split if its vertex set can be partitioned into a clique and an independent set. A split graph is $(x,y)$-bidegreed if each of its vertex degrees is equal to either $x$ or $y$. Each connected split graph is of diameter at most 3. In 2017, Nikiforov proposed the $A_{\alpha }$-matrix, which is the convex combination of the adjacency matrix and the diagonal matrix of vertex degrees of the graph $G$. It is well-known that a connected graph of diameter $l$ contains at least $l+1$ distinct $A_{\alpha }$-eigenvalues. A graph is said to be $l_{\alpha }$-extremal with respect to its $A_{\alpha }$-matrix if the graph is of diameter $l$ having exactly $l+1$ distinct $A_{\alpha }$-eigenvalues. In this paper, using the association of split graphs with combinatorial designs, the connected $2_{\alpha }$-extremal (resp. $3_{\alpha }$-extremal) bidegreed split graphs are classified. Furthermore, all connected bidegreed split graphs of diameter 2 having just 4 distinct $A_{\alpha }$-eigenvalues are identified.