The existence of a $\{P_2,C_3,P_5,\mathcal{T}(3)\}$-factor based on the size or the $A_{\alpha }$-spectral radius of graphs

Let $G$ be a connected graph of order $n$. A $\{P_2,C_3,P_5,\mathcal{T}(3)\}$-factor of $G$ is a spanning subgraph of $G$ such that each component is isomorphic to a member in $\{P_2,C_3,P_5,\mathcal{T}(3)\}$, where $\mathcal{T}(3)$ is a $\{1,2,3\}$-tree. The $A_{\alpha }$-spectral radius of $G$ is denoted by ${\rho }_{\alpha }(G)$. In this paper, we obtain a lower bound on the size or the $A_{\alpha }$-spectral radius for $\alpha \in [0,1)$ of $G$ to guarantee that $G$ has a $\{P_2,C_3,P_5,\mathcal{T}(3)\}$-factor, and construct an extremal graph to show that the bound on $A_{\alpha }$-spectral radius is optimal.