Star-factors with large components, fractional k-extendability and spectral radius in graphs

Abstract

Let $G$ be a graph, and let $m$ and $k$ be two integers with $m\ge 2$ and $k\ge 1$. A $\{K_{1,j}:m\le j\le 2m\}$-factor of $G$ is a spanning subgraph of $G$, in which every component is isomorphic to a member in $\{K_{1,j}:m\le j\le 2m\}$. A graph $G$ is fractional $k$-extendable if every $k$-matching in $G$ can be extended to a fractional perfect matching of $G$. In this paper, we first establish a lower bound on the $A_a$-spectral radius of $G$ to guarantee that $G$ has a $\{K_{1,j}:m\le j\le 2m\}$-factor, where $a\in \{0,1\}$. Then we determine a lower bound on the $A_{\alpha }$-spectral radius of $G$ to ensure that $G$ is fractional $k$-extendable, where $\alpha \in [0,1)$.