The existence of a {P2, C3, P5, T (3)}-factor based on the size or the Aα-spectral radius of graphs

Let G be a connected graph of order n. A {P2, C3, P5, T (3)}-factor of G is a spanning subgraph of G such that each component is isomorphic to a member in {P2, C3, P5, T (3)}, where T (3) is a {1, 2, 3}-tree. The A?-spectral radius of G is denoted by ?? (G). In this paper, we obtain a lower bound on the size or the A?-spectral radius for ? ? [0, 1) of G to guarantee that G has a {P2, C3, P5, T (3)}-factor, and construct an extremal graph to show that the bound on A?-spectral radius is optimal.