The Aα-spectral radius for path-factors in graphs

Let alpha alpha is an element of [0,1), and let G be a connected graph of order n with n >= f (alpha), where f(alpha)=14 for alpha is an element of [0,12], f(alpha)=17 for alpha is an element of (12,23], f(alpha)=20 for alpha is an element of (23,34] and f(alpha)=51-alpha+1 for alpha is an element of (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 A alpha-spectral radius of G is denoted by rho alpha(G). In this paper, it is verified that G has a P >= 2-factor if rho alpha(G)>theta(n), where theta(n) is the largest root of x(3)-((alpha+1)n+alpha-5)x(2)+(alpha n(2)+(alpha(2)-3 alpha-1)n-2 alpha+1)x-alpha(2)n(2+)(7 alpha(2)-5 alpha+3)n-18 alpha(2)+29 alpha-15=0. Furthermore, we provide a graph to show that the bound on A alpha-spectral radiusis optimal. (c) 2024 Elsevier B.V. All rights reserved