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

TL;DR

This paper establishes spectral radius conditions based on distance matrices that guarantee the existence of perfect $k$-matchings, generalized factor-criticality, and $k$-$d$-criticality in graphs, advancing spectral graph theory.

Contribution

It introduces new spectral radius criteria involving distance matrices that ensure complex matching and criticality properties in graphs, extending prior spectral conditions.

Findings

Spectral radius conditions guarantee perfect $k$-matchings.

Conditions ensure graphs are $k$-$d$-critical, GFC$_k$, or GBC$_k$.

Provides sufficient spectral criteria for advanced graph properties.

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.