Spectral radius, toughness and $k$-factor of graphs

A $k$-regular spanning subgraph of $G$ is called a $k$-factor. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] presented a tight sufficient condition in terms of the spectral radius for a connected 1-tough graph to contain a connected 2-factor (Hamilton cycle). Then it is interesting to consider the following problem: What is the spectral radius condition to guarantee the existence of a $k$-factor with $k\ge3$ in a connected 1-tough graph $G$ with $\delta(G)\ge k$? In this paper, we completely solve this problem.