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 $f:E(G)\to \{0,1,\dots ,k\}$ 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: \[ \defk(G) = \max_{S \subseteq V(G)} \begin{cases} k \cdot i(G - S) - k|S|, & k \text{ is even;} \\[6pt] \odd(G - S) + k \cdot i(G - S) - k|S|, & k \text{ is odd.} \end{cases} \] 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 $\varnothing$ 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 ${\sum }_{e\in E_G(v)}h(e)=k-d$ {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.