Quantum motivation
Noncontextuality inequalities (and in particular Bell inequalities) can be mapped into graphs, in such a way that its relevant properties can be calculated via some simple graph-theoretical functions. In particular, the maximal quantum violation of a noncontextuality inequality is given by the Lovász function of its graph.
A family of inequalities that interests myself particularly (is it appropriate to cite my own paper here?) is represented by the Möbius ladder; I was unable to find its Lovász function in the literature, but it can be proven in a somewhat tortuous way from some well-known results in physics.
Quantum-free question
Let $M_{2n}$ be the Möbius ladder graph. Its Lovász function is
$$\vartheta(M_{2n}) = \frac{n}{2}\bigg(1+\cos\frac{\pi}{n}\bigg).$$ I lack, however, a direct proof of this fact. Is there one?
