Sufficient conditions for fractional k-extendable graphs

Let k and n be two positive integers. A graph G is said to be fractional k-extendable for 0 ? k ?n-22 if every k-matching M in G is contained in a fractional perfect matching G[Fh] of G such that h(e) = 1 for all e ? M, where h: E(G) ? [0, 1] be a function. Let e(G) denote the size of G and ?(G) denote the spectral radius of G. In this paper, we first provide a tight size condition to ensure that a connected graph is fractional k-extendable. Then, we determine a lower bound on the spectral radius of a connected graph G to guarantee that G is fractional k-extendable. Finally, we construct some extremal graphs to show that all the bounds are sharp.