On the distance signless Laplacian spectral radius, fractional matching and factors of graphs
Z. H. Zhang, L. G. Wang
TL;DR
The work links the distance signless Laplacian spectral radius $\eta(G)$ to fractional matchings and graph factors. It provides a tight upper bound on $\eta(G)$ that guarantees $\mu_f(G) > \frac{n-k}{2}$, using equitable partitions and spectral comparisons against a specific extremal graph. It also establishes a tight spectral-condition guaranteeing the existence of a $\{K_2,\{C_k\}\}$-factor for $k\ge3$, with explicit small-$n$ exceptions. Together, these results furnish spectral criteria for key combinatorial structures in graphs and highlight a precise extremal boundary via $K_1\vee(K_{n-2-k}+\overline{K_{1+k}})$.
Abstract
The distance signless Laplacian matrix of a graph $G$ is define as $Q(G)=$Tr$(G)+D(G)$, where Tr$(G)$ and $D(G)$ are the diagonal matrix of vertex transmissions and the distance matrix of $G$, respectively. Denote by $E_G(v)$ the set of all edges incident to a vertex $v$ in $G$. A fractional matching of a graph $G$ is a function $f:E(G) \rightarrow [0,1]$ such that $\sum_{e\in E_G(v)} f(e)\leq 1$ for every vertex $v\in V(G)$. The fractional matching number $μ_f(G)$ of a graph $G$ is the maximum value of $ \sum_{e\in E(G)} f(e)$ over all fractional matchings. Given subgraphs $H_1, H_2,...,H_k$ of $G$, a $\{H_1, H_2,...,H_k\}$-factor of $G$ is a spanning subgraph $F$ in which each connected component is isomorphic to one of $H_1, H_2,...,H_k$. In this paper, we establish a upper bound for the distance signless Laplacian spectral radius of a graph $G$ of order $n$ to guarantee that $μ_f(G)> \frac{n-k}{2}$, where $1\leq k<n$ is an integer. Besides, we also provide a sufficient condition based on distance signless Laplacian spectral radius to guarantee the existence of a $\{K_2,\{C_k\}\}$-factor in a graph, where $k \geq 3$ is an integer.