Computing the Grothendieck constant of some graph classes

The Grothendieck constant $\kappa \left(G\right)$ of a graph $G=(\left[n\right],E)$ is the integrality gap of the canonical semidefinite relaxation of the integer program ${max}_{x\in {\{\pm 1\}}^n}{\Sigma }_{ij\in E}{w}_{ij}{x}_i·x_j$ , replacing $\pm 1$ variables by unit vectors. We show that $\kappa \left(G\right)=g/(g-2)cos(\pi /g)\le 3/2$ when $G$ has no $K_5$ ‐minor and girth $g$ ; moreover, $\kappa \left(G\right)\le \kappa \left(K_k\right)$ if the cut polytope of $G$ is defined by inequalities supported by at most $k$ points; lastly the worst case ratio of clique‐web inequalities is bounded by 3.