Size and spectral radius conditions for the existence of $k$-factor in graphs with given minimum degree

In recent years, characterizing the conditions for the existence of an $[a,b]$-factor in graphs has received significant academic attention. For integers $b\ge a\ge 1$, an $[a,b]$-factor of a graph $G$ is defined as a spanning subgraph $H$ such that $a\le d_H(v)\le b$ for every $v\in V(G)$. If $a=b=k$, a $[k,k]$-factor is called a $k$-factor of $G$. Let $\delta (G)$ denote the minimum degree of a graph $G$. Hao and Li [Electron. J. Combin. (2024)] posed the problem of characterizing the size or spectral radius conditions for a graph to contain an $[a,b]$-factor with $\delta (G)\ge a$. Fan, Lin and Zhu [Discrete Math. (2026)] solved the size version of this problem for the case $a<b\mathrm{.}$ In this paper, for $a=b=k$, we characterize the size condition for a graph to have a $k$-factor with $\delta (G)\ge k$. Using this result, we also characterize the spectral radius and distance spectral radius conditions for a graph to have a $k$-factor, respectively.