The difference and ratio of the fractional matching number and the matching number of graphs

Given a graph GG, the matching number   of GG, written α′(G)α′(G), is the maximum size of a matching in GG, and the fractional matching number   of GG, written αf′(G), is the maximum size of a fractional matching of GG. In this paper, we prove that if GG is an nn-vertex connected graph that is neither K1K1 nor K3K3, then αf′(G)−α′(G)≤n−26 and αf′(G)α′(G)≤3n2n+2. Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold.