Some spectral conditions for star-factors in bipartite graphs

A spanning subgraph $F$ of $G$ is called an $F$-factor if every component of $F$ is isomorphic to some member of $F$, where $F$ is a set of connected graphs. We denote by $A\left(G\right)$ the adjacency matrix of $G$, and by $D\left(G\right)$ the distance matrix of $G$. The largest eigenvalue of $A\left(G\right)$, denoted by $\rho \left(G\right)$, is called the adjacency spectral radius of $G$. The largest eigenvalue of $D\left(G\right)$, denoted by $\mu \left(G\right)$, is called the distance spectral radius of $G$. In this paper, we aim to provide two spectral conditions to ensure the existence of star-factors with given properties. Let $G$ be a $k$-edge-connected bipartite graph with bipartition $(A,B)$ and $\left|B\right|=k\left|A\right|=kn$, where $n$ is a sufficiently large positive integer. Then the following two results are true.

(i) If $\rho \left(G\right)\ge \rho (K_{n-1,kn-k-1}\nabla K_{1,k+1})$, then $G$ contains a star-factor $F$ with $d_F\left(u\right)=k$ for any $u\in A$ and $d_F\left(v\right)=1$ for any $v\in B$, unless $G=K_{n-1,kn-k-1}\nabla K_{1,k+1}$.

(ii) If $\mu \left(G\right)\le \mu (K_{n-1,kn-k-1}\nabla K_{1,k+1})$, then $G$ contains a star-factor $F$ with $d_F\left(u\right)=k$ for any $u\in A$ and $d_F\left(v\right)=1$ for any $v\in B$, unless $G=K_{n-1,kn-k-1}\nabla K_{1,k+1}$.