  Let $a$ and $b$ be two positive integers with $a\leq b$, and let $G$ be a
graph with vertex set $V(G)$ and edge set $E(G)$. Let $h:E(G)\rightarrow[0,1]$
be a function. If $a\leq\sum\limits_{e\in E_G(v)}{h(e)}\leq b$ holds for every
$v\in V(G)$, then the subgraph of $G$ with vertex set $V(G)$ and edge set
$F_h$, denoted by $G[F_h]$, is called a fractional $[a,b]$-factor of $G$ with
indicator function $h$, where $E_G(v)$ denotes the set of edges incident with
$v$ in $G$ and $F_h=\{e\in E(G):h(e)>0\}$. A graph $G$ is defined as a
fractional $[a,b]$-deleted graph if for any $e\in E(G)$, $G-e$ contains a
fractional $[a,b]$-factor. The size, spectral radius and signless Laplacian
spectral radius of $G$ are denoted by $e(G)$, $\rho(G)$ and $q(G)$,
respectively. In this paper, we establish a lower bound on the size, spectral
radius and signless Laplacian spectral radius of a graph $G$ to guarantee that
$G$ is a fractional $[a,b]$-deleted graph.
