October 2023 COMPLETE CHARACTERIZATION OF THE BIDEGREED SPLIT GRAPHS WITH THREE OR FOUR DISTINCT Aα-EIGENVALUES
Wanting Sun, Shuchao Li, Xuechao Li
Rocky Mountain J. Math. 53(5): 1571-1585 (October 2023). DOI: 10.1216/rmj.2023.53.1571

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.

Citation

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Wanting Sun. Shuchao Li. Xuechao Li. "COMPLETE CHARACTERIZATION OF THE BIDEGREED SPLIT GRAPHS WITH THREE OR FOUR DISTINCT Aα-EIGENVALUES." Rocky Mountain J. Math. 53 (5) 1571 - 1585, October 2023. https://doi.org/10.1216/rmj.2023.53.1571

Information

Received: 15 December 2021; Revised: 10 April 2022; Accepted: 17 October 2022; Published: October 2023
First available in Project Euclid: 19 September 2023

MathSciNet: MR4643820
Digital Object Identifier: 10.1216/rmj.2023.53.1571

Subjects:
Primary: 05C50
Secondary: 05C75

Keywords: Aα-eigenvalue , Aα-matrix , combinatorial design , split graph

Rights: Copyright © 2023 Rocky Mountain Mathematics Consortium

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Vol.53 • No. 5 • October 2023
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