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.
Categories: math.CO