Aα-spectral radius and path-factor covered graphs

Let α∈[0,1), and let G be a connected graph of order n with n≥ f(α), where f(α)=14 for α∈[0,1/2], f(α)=17 for α∈(1/2,2/3], f(α)=20 for α∈(2/3,3/4] and f(α)=5/1-α+1 for α∈(3/4,1). A path factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. Let k≥2 be an integer. A P_≥ k-factor means a path-factor with each component being a path of order at least k. A graph G is called a P_≥ k-factor covered graph if G has a P_≥ k-factor containing e for any e∈ E(G). Let A_α(G)=α D(G)+(1-α)A(G), where D(G) denotes the diagonal matrix of vertex degrees of G and A(G) denotes the adjacency matrix of G. The largest eigenvalue of A_α(G) is called the A_α-spectral radius of G, which is denoted by ρ_α(G). In this paper, it is proved that G is a P_≥2-factor covered graph if ρ_α(G)>η(n), where η(n) is the largest root of x^3-((α+1)n+α-4)x^2+(α n^2+(α^2-2α-1)n-2α+1)x-α^2n^2+(5α^2-3α+2)n-10α^2+15α-8=0. Furthermore, we provide a graph to show that the bound on A_α-spectral radius is optimal.