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

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 $f:E(G)\to [0,1]$ such that ${\sum }_{e\in E_G(v)}f(e)\le 1$ 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 $1\le k<n$ 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 $k\ge 3$ is an integer.

10 pages