  We relate homological properties of a binomial edge ideal $\mathcal{J}_G$ to
invariants that measure the connectivity of a simple graph $G$. Specifically,
we show if $R/\mathcal{J}_G$ is a Cohen-Macaulay ring, then graph toughness of
$G$ is exactly $\frac{1}{2}$. We also give an inequality between the depth of
$R/\mathcal{J}_G$ and the vertex-connectivity of $G$. In addition, we study the
Hilbert-Samuel multiplicity, and the Hilbert-Kunz multiplicity of
$R/\mathcal{J}_G$.
