The Aα-spectral radius for path-factors in graphs

• Published in: Discrete mathematics Vol. 347; no. 5
• Main Authors: Zhou, Sizhong, Zhang, Yuli, Sun, Zhiren
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.05.2024
• Subjects: [formula omitted]-factor; [formula omitted]-spectral radius; Graph; Aα-spectral radius; Graph; P≥2-factor
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2024.113940

Let α∈[0,1), and let G be a connected graph of order n with n≥f(α), where f(α)=14 for α∈[0,12], f(α)=17 for α∈(12,23], f(α)=20 for α∈(23,34] and f(α)=51−α+1 for α∈(34,1). A spanning subgraph whose components are paths is said to be a path-factor. A P≥k-factor means a path-factor with each component being a path of order at least k, where k≥2 is an integer. The Aα-spectral radius of G is denoted by ρα(G). In this paper, it is verified that G has a P≥2-factor if ρα(G)>θ(n), where θ(n) is the largest root of x3−((α+1)n+α−5)x2+(αn2+(α2−3α−1)n−2α+1)x−α2n2+(7α2−5α+3)n−18α2+29α−15=0. Furthermore, we provide a graph to show that the bound on Aα-spectral radius is optimal.