On the distance signless Laplacian spectral radius, fractional matching and factors of graphs

TL;DR

This paper investigates the relationship between the distance signless Laplacian spectral radius of a graph and its fractional matchings and factors, providing bounds and conditions for their existence.

Contribution

It establishes new upper bounds on the spectral radius to guarantee fractional matchings and factors, advancing understanding of spectral conditions for graph properties.

Findings

Upper bound for spectral radius ensuring fractional matching number > (n-k)/2

Spectral radius condition guarantees existence of {K_2, C_k}-factor

Provides sufficient spectral conditions for specific graph factors

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 $\mu_f(G)$ 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 $\mu_f(G)> \frac{n-k}{2}$, 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.