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 Aα-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 Aα-eigenvalues. A graph is said to be lα-extremal with respect to its Aα-matrix if the graph is of diameter l having exactly l+1 distinct Aα-eigenvalues. In this paper, using the association of split graphs with combinatorial designs, the connected 2α-extremal (resp. 3α-extremal) bidegreed split graphs are classified. Furthermore, all connected bidegreed split graphs of diameter 2 having just 4 distinct Aα-eigenvalues are identified.