The existence of a {K1,2,K1,3,K5}-factor based on the size or the Aα-spectral radius of graphs

Let $G$ be a connected graph of order $n$. A $\{K_{1,2},K_{1,3},K_5\}$-factor of $G$ is a spanning subgraph of $G$, in which each component is isomorphic to a member in $\{K_{1,2},K_{1,3},K_5\}$. The $A_{\alpha }$-spectral radius of $G$ is denoted by ${\rho }_{\alpha }\left(G\right)$. In this paper, we obtain a lower bound on the size (resp. the $A_{\alpha }$-spectral radius for $\alpha \in [0,\frac{1}{2}]$) of $G$ to guarantee that $G$ has a $\{K_{1,2},K_{1,3},K_5\}$-factor, and show that $G$ has a $\{K_{1,2},K_{1,3},K_5\}$-factor if ${\rho }_{\alpha }\left(G\right)>\tau \left(n\right)$, where $\tau \left(n\right)$ is the largest root of $x^3-((\alpha +1)n+\alpha -3)x^2+(\alpha n^2+({\alpha }^2-\alpha -1)n-2\alpha +1)x-{\alpha }^2{n}^2+(3{\alpha }^2-\alpha +1)n-4{\alpha }^2+5\alpha -3=0$.