  Given a graph $G=([n],E)$ and $w\in\R^E$, consider the integer program
${\max}_{x\in \{\pm 1\}^n} \sum_{ij \in E} w_{ij}x_ix_j$ and its canonical
semidefinite programming relaxation ${\max} \sum_{ij \in E} w_{ij}v_i^Tv_j$,
where the maximum is taken over all unit vectors $v_i\in\R^n$. The integrality
gap of this relaxation is known as the Grothendieck constant $\ka(G)$ of $G$.
We present a closed-form formula for the Grothendieck constant of $K_5$-minor
free graphs and derive that it is at most 3/2. Moreover, we show that
$\ka(G)\le \ka(K_k)$ if the cut polytope of $G$ is defined by inequalities
supported by at most $k$ points. Lastly, since the Grothendieck constant of
$K_n$ grows as $\Theta(\log n)$, it is interesting to identify instances with
large gap. However this is not the case for the clique-web inequalities, a wide
class of valid inequalities for the cut polytope, whose integrality ratio is
shown to be bounded by 3.
