For $\alpha \ge 2$, an $\alpha$-vector coloring of a graph $X$ is an assignment of unit vectors to the vertices of $X$ such that vectors assigned to adjacent vertices have inner product less than or equal to $\frac{1}{1-\alpha}$. A strict $\alpha$-vector coloring requires equality for the inner product.

The vector chromatic number of a graph $X$, denoted $\chi_{vec}(X)$, is defined to be the minimum $\alpha$ for which $X$ has an $\alpha$-vector coloring. Strict vector chromatic number is defined analogously. It is known that the strict vector chromatic number of $X$ is equal to the Lovasz theta function of the complement of $X$.

**Are there any finite graphs known for which the vector and strict vector chromatic numbers are different?**

For reference, vector and strict vector colorings and the relation to Lovasz theta come from "Approximate Graph Coloring by Semidefinite Programming" by Karger, Motwani, and Sudan.