完美k-匹配的距离谱半径条件，广义因数-临界性（双临界性）和k-d临界性的图
Distance spectral radius conditions for perfect $k$-matching, generalized factor-criticality (bicriticality) and $k$-$d$-criticality of graphs
AI总结：
本文通过距离谱半径条件，研究了图完美k-匹配、k-d临界性、广义因数-临界性和双临界性等性质的充分条件。
摘要：
设G为一个简单连通图，其顶点集为V(G)，边集为E(G)。图G的k-匹配是一个函数f:E(G)→{0,1,…,k}，满足对于每个顶点v∈V(G)，∑_{e∈E_G(v)} f(e) ≤k。如果对于任意顶点v∈V(G)，∑_{e∈E_G(v)} f(e)=k，则称该匹配为完美匹配。图G的k-Berge-Tutte公式定义为：\n$$\ndefk(G) = \max_{S ⊆ V(G)}\begin{cases}k·i(G - S) -k|S|, &k为偶数；\[6pt]\odd(G - S) +k·i(G - S) -k|S|, &k为奇数。\end{cases}$$\n图G的k-屏障是使k-Berge-Tutte公式达到最大值的顶点子集S。一个奇（偶）阶的连通图G被称为关于整数k-匹配的广义因数-临界（广义双临界）图，简称为GFC_k（GBC_k）图，如果∅是唯一的k-屏障。当k为奇数时，令1≤d≤k且|V(G)|≡d mod 2。如果对于任意v∈V(G)，存在一个k-匹配h，使得∑_{e∈E_G(v)} h(e)=k-d且对于任意u∈V(G)-{v}，∑_{e∈E_G(u)} h(e)=k，则称G为k-d临界图。本文中，我们提供了基于距离谱半径的充分条件，以确保图具有完美k-匹配，图是k-d临界图、GFC_k图或GBC_k图。
英文摘要：
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 $\sum_{e \in E_G(v)} h(e) = k - d$ {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.
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