On the distance spectral radius, fractional matching and factors of graphs with given minimum degree

Abstract

A fractional matching of $G$ is a function $f: E(G)\to [0,1]$ such that $\sum_{e\in E_G(v_i)}f(e)\le 1$ for any $v_i\in V(G)$, where $E_G(v_i)=\{e: e\in E(G) \ \textrm{and}\ e \ \textrm{is incident with} \ v_i\}$. Let $\alpha_f(G)$ denote the fractional matching number of $G$, which is defined as $\alpha_f(G)=\max\{\sum_{e\in E(G)}f(e): f\ \textrm{is a fractional matching of} \ G\}$. Let $\{G_1,G_2,G_3,\dots\}$ be a set of graphs, a $\{G_1,G_2,G_3,\dots\}$-factor of a graph $G$ is a spanning subgraph of $G$ such that each component of which is isomorphic to one of $\{G_1,G_2,G_3,\dots\}$. In this paper, we first establish a sharp upper bound for the distance spectral radius to guarantee that $\alpha_f(G)>\frac{n-k}{2}$ in a graph $G$ of order $n$ with given minimum degree, where $0<k<n$ is an integer. Then we give a sharp upper bound on the distance spectral radius of a graph $G$ with given minimum degree $\delta$ to ensure that $G$ has a $\{K_2, \{C_k\}\}$-factor, where $3\le k<+\infty$ is an integer. Moreover, we obtain a sharp upper bound on the distance spectral radius for the existence of a $\{K_{1,1},K_{1,2},\dots,K_{1,k}\}$-factor with $2\le k<+\infty$ in a graph $G$ with given minimum degree.