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

Given a graph $G$, the matching number of $G$, written ${\alpha }^{\mathrm{\prime }}(G)$, is the maximum size of a matching in $G$, and the fractional matching number of $G$, written ${\alpha }_f^{\mathrm{\prime }}(G)$, is the maximum size of a fractional matching of $G$. In this paper, we prove that if $G$ is an $n$-vertex connected graph that is neither $K_1$ nor $K_3$, then ${\alpha }_f^{\mathrm{\prime }}(G)-{\alpha }^{\mathrm{\prime }}(G)\le \frac{n-2}{6}$ and $\frac{{\alpha }_f^{\mathrm{\prime }}(G)}{{\alpha }^{\mathrm{\prime }}(G)}\le \frac{3n}{2n+2}$. Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold.