Spectral radius and fractional [a,b]-factor of graphs

Abstract

Let $h:E\left(G\right)\to [0,1]$ be a function on $E\left(G\right)$ and let $a,b$ be two positive integers with $a\le b$. If $a\le {\sum }_{e\in E_G\left(v\right)}h\left(e\right)\le b$ for any $v\in V\left(G\right)$, then the spanning subgraph with edge set $E_h=\{e\in E(G\left)\right|h\left(e\right)>0\}$, denoted by $G\left[E_h\right]$, is called a fractional $[a,b]$-factor of G with indicator function h. In this paper, we provide a spectral condition to guarantee the existence of a fractional $[a,b]$-factor in a graph with minimum degree $\delta \ge a\ge 1$, which extends some previous results. Moreover, we also provide a lower bound on the size of a graph to guarantee the existence of a fractional $[a,b]$-factor for $b\ge a\ge 1$.