Distance spectral radius conditions for perfect $k$-matching, generalized factor-criticality (bicriticality) and $k$-$d$-criticality of graphs
Yang kexin, Wang ligong, Zhang zhenhao
TL;DR
This work links the distance spectral radius $\lambda_1(D(G))$ to generalized matching properties in graphs. By leveraging the $k$-Berge-Tutte framework, barrier structure, and precise graph-structure decompositions, the authors derive explicit spectral-radius thresholds, relative to extremal graphs such as $S_{n,k}$, $G^s$, and $K_1\vee(K_{n-2}\cup K_1)$, that guarantee the existence of a perfect $k$-matching and, for odd $k$, $k$-$d$-criticality, as well as $\mathrm{GFC}_k$ and $\mathrm{GBC}_k$ properties for even $k$. The results extend prior adjacency- and fractional-matching spectral criteria to distance-based criteria, offering tight, parity-aware conditions and identifying extremal graphs where the bounds are achieved. These criteria have potential implications for graph design and verification of generalized matching properties via spectral data.
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 $\sum_{e \in E_G(v)} h(e) = k - d$ {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.