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

作者：Sun, Wanting; Li, Shuchao^*; Li, Xuechao

来源：ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, 2023, 53(5): 1571-1585.

DOI：10.1216/rmj.2023.53.1571

• split graph
• A(alpha)-matrix
• A(alpha)-eigenvalue
• combinatorial design

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.

• 单位
  山东大学