Sufficient conditions for fractional [a, b]-deleted graphs

Let a and b be two positive integers with a <= b, and let G be a graph with vertex set V(G)and edge set E(G).Let h:E(G)->[0,1]be a function. If a <=& sum;e is an element of EG(v)h(e)<= b holds for every v is an element of V(G), then the subgraph of G with vertex set V(G)and edge set Fh, denoted by G[Fh],is called a fractional[a,b]-factor of G within dicator function h, where EG(v)denotes the set of edges incident with v in G and Fh={e is an element of E(G):h(e)>0}. A graph G is defined as a fractional[a,b]-deleted graph if for any e is an element of E(G),G-e contains a fractional[a,b]-factor. The size, spectral radius and signless Laplacian spectral radius of Gare denoted bye(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