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

Abstract: 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 $f:E(G)\rightarrow {0,1,\ldots, k}$ satisfying $\sum_{e \in E_G(v)} f(e) \leq k$ 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: [ \defk(G) = \max_{S \subseteq V(G)} \begin{cases} k \cdot i(G - S) - k|S|, & k \text{ is even;} \[6pt] \odd(G - S) + k \cdot i(G - S) - k|S|, & k \text{ is odd.} \end{cases} ] 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{GFC}k (\mathrm{GBC}_k)) graph, if $\emptyset$ is a unique $k$-barrier. When $k$ is odd, let ( 1 \leq d \leq k ) and ( |V(G)| \equiv d \pmod{2} ). If for any ( v \in V(G) ), there exists a ( k )-matching ( h ) such that $\sum</em>{e \in E_G(v)} h(e) = k - d$ {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, $\mathrm{GFC}_k$ or $\mathrm{GBC}_k$, respectively.