Spectral conditions for graphs to contain $k$-factors

Abstract

Let $G$ be a graph. The spectral radius $\rho (G)$ of $G$ is the largest eigenvalue of its adjacency matrix. For an integer $k\ge 1$, a $k$-factor of $G$ is a $k$-regular spanning subgraph of $G$. Assume that $k$ and $n$ are integers satisfying $k\ge 2,kn\equiv 0(\mathrm{mod}2)$ and $n\ge \max \left\{k^2+6k+7,20k+10\right\}$. Let $G$ be a graph of order $n$ and with minimum degree at least $k$. In this paper, we give a sharp lower bound of $\rho (G)$ to guarantee that $G$ contains a $k$-factor.