An $A_{\alpha }$-spectral radius for the existence of $\{P_3,P_4,P_5\}$-factors in graphs

Let $G$ be a connected graph of order $n$ with $n\ge 25$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. Nikiforov introduced the $A_{\alpha }$-matrix of $G$ as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$ [V. Nikiforov, Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math., 11 (2017), 81–107], where $\alpha \in [0,1]$, $D(G)$ denotes the diagonal matrix of vertex degrees of $G$ and $A(G)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_{\alpha }(G)$, denoted by ${\lambda }_{\alpha }(G)$, is called the $A_{\alpha }$-spectral radius of $G$. In this paper, it is proved that $G$ has a $\{P_3,P_4,P_5\}$-factor unless $G=K_1\vee (K_{n-2}\cup K_1)$ if ${\lambda }_{\alpha }(G)\ge {\lambda }_{\alpha }(K_1\vee (K_{n-2}\cup K_1))$, where $\alpha$ is a real number with $0\le \alpha <\frac{2}{3}$.