Let a and b be two positive integers with \(a\le b\) , 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 \(a\le \sum \limits _{e\in E_G(v)}{h(e)}\le b\) 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.
