$ A_{\alpha} $-spectral radius and path-factor covered graphs

Let $ \alpha\in[0, 1) $, and let $ G $ be a connected graph of order $ n $ with $ n\geq f(\alpha) $, where $ f(\alpha) = 14 $ for $ \alpha\in[0, \frac{1}{2}] $, $ f(\alpha) = 17 $ for $ \alpha\in(\frac{1}{2}, \frac{2}{3}] $, $ f(\alpha) = 20 $ for $ \alpha\in(\frac{2}{3}, \frac{3}{4}] $ and $ f(\alpha) = \frac{5}{1-\alpha}+1 $ for $ \alpha\in(\frac{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\geq2 $ be an integer. A $ P_{\geq k} $-factor means a path-factor with each component being a path of order at least $ k $. A graph $ G $ is called a $ P_{\geq k} $-factor covered graph if $ G $ has a $ P_{\geq k} $-factor containing $ e $ for any $ e\in E(G) $. Let $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha)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_{\alpha}(G) $ is called the $ A_{\alpha} $-spectral radius of $ G $, which is denoted by $ \rho_{\alpha}(G) $. In this paper, it is proved that $ G $ is a $ P_{\geq2} $-factor covered graph if $ \rho_{\alpha}(G) > \eta(n) $, where $ \eta(n) $ is the largest root of $ x^{3}-((\alpha+1)n+\alpha-4)x^{2}+(\alpha n^{2}+(\alpha^{2}-2\alpha-1)n-2\alpha+1)x-\alpha^{2}n^{2}+(5\alpha^{2}-3\alpha+2)n -10\alpha^{2}+15\alpha-8 = 0 $. Furthermore, we provide a graph to show that the bound on $ A_{\alpha} $-spectral radius is optimal.