A{α}-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 Aα(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 Aα(G) is called the Aα-spectral radius of G, which is denoted by ρα(G). In this paper, it is proved that G is a P≥2-factor covered graph if ρα(G)>η(n), where η(n) is the largest root of x³-((α+1)n+α-4)x²+(α n²+(α²-2α-1)n-2α+1)x-α²n²+(5α²-3α+2)n-10α²+15α-8=0. Furthermore, we provide a graph to show that the bound on Aα-spectral radius is optimal.