Distance spectral radius conditions for perfect $k$-matching, generalized factor-criticality (bicriticality) and $k$-$d$-criticality of graphs

Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-matching of a graph $G$ is a function $f:E(G)\to \{0,1,\dots ,k\}$ satisfying ${\sum }_{e\in E_G(v)}f(e)\le k$ for every vertex $v\in V(G)$, where $E_G(v)$ is the set of edges incident with $v$ in $G$. A $k$-matching of a graph $G$ is perfect if ${\sum }_{e\in E_G(v)}f(e)=k$ for any vertex $v\in V(G)$. The $k$-Berge-Tutte-formula of a graph $G$ is defined as: $$\text{defk}(G)=\underset{S\subseteq V(G)}{\max }\{\begin{array}{ll}k\cdot i(G-S)-k\mathrm{\mid }S\mathrm{\mid },& k\text{ is even;}\\ \text{odd}(G-S)+k\cdot i(G-S)-k\mathrm{\mid }S\mathrm{\mid },& k\text{ is odd.}\end{array}$$ A $k$-barrier of the graph $G$ is the subset $S\subseteq V(G)$ that reaches the maximum value in $k$-Berge-Tutte-formula. A connected graph $G$ of odd (even) order is a {generalized factor-critical (generalized bicritical) graph about integer $k$-matching}, abbreviated as a ${\mathrm{G}\mathrm{F}\mathrm{C}}_k({\mathrm{G}\mathrm{B}\mathrm{C}}_k)$ graph, if $\mathrm{\varnothing }$ is a unique $k$-barrier. When $k$ is odd, let $1\le d\le k$ and $\mathrm{\mid }V(G)\mathrm{\mid }\equiv d(\mathrm{m}\mathrm{o}\mathrm{d}2)$. If for any $v\in V(G)$, there exists a $k$-matching $h$ such that ${\sum }_{e\in E_G(v)}h(e)=k-d$ {and} ${\sum }_{e\in E_G(u)}h(e)=k$ for any $u\in V(G)-\{v\}$, then $G$ is said to be $k$-$d$-critical. In this paper, we provide sufficient conditions in terms of distance spectral radius to ensure that a graph has a perfect $k$-matching and a graph is $k$-$d$-critical, ${\mathrm{G}\mathrm{F}\mathrm{C}}_k$ or ${\mathrm{G}\mathrm{B}\mathrm{C}}_k$, respectively.