Graphs determined by their Aα-spectra

• Published in: Discrete mathematics Vol. 342; no. 2; pp. 441 - 450
• Main Authors: Lin, Huiqiu, Liu, Xiaogang, Xue, Jie
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2019
• Subjects: [formula omitted]-spectrum; Determined by the [formula omitted]-spectrum; Join; Aα-spectrum; Join; Determined by the Aα-spectrum
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2018.10.006

Let G be a graph with n vertices, and let A(G) and D(G) denote respectively the adjacency matrix and the degree matrix of G. Define Aα(G)=αD(G)+(1−α)A(G)for any real α∈[0,1]. The collection of eigenvalues of Aα(G) together with multiplicities are called the Aα-spectrum of G. A graph G is said to be determined by itsAα-spectrum if all graphs having the same Aα-spectrum as G are isomorphic to G. We first prove that some graphs are determined by their Aα-spectra for 0≤α<1, including the complete graph Kn, the union of cycles, the complement of the union of cycles, the union of copies of K2 and K1, the complement of the union of copies of K2 and K1, the path Pn, and the complement of Pn. Setting α=0 or 12, those graphs are determined by A- or Q-spectra. Secondly, when G is regular, we show that G is determined by its Aα-spectrum if and only if the join G∨Km (m≥2) is determined by its Aα-spectrum for 12<α<1. Furthermore, we also show that the join Km∨Pn (m,n≥2) is determined by its Aα-spectrum for 12<α<1. In the end, we pose some related open problems for future study.