Lovasz theta of even cycle How does one show Lovasz theta of even $n$-cycle ($n$ is even) is of form $\frac{n}{2}$? Why is the Lovasz theta of such cycles not of form $\frac{n \cos(\frac{\pi}{n})}{1+\cos(\frac{\pi}{n})}$. Could someone provide a derivation for even cycle Lovasz theta number. It is clear that the Shannon Capacity is $\frac{n}{2}$. why is the cosine form tight only for odd cycles?
The Lovász theta function is bounded between the independence number and the clique covering number (the chromatic number of the complementary graph). For even cycles, both numbers are $n/2$. For example, for the $6$-cycle with vertices $1,2,3,4,5,6$, there are independent sets of size $3$, e.g. $\{1,3,5\}$, and the graph can be covered with the $3$ cliques $\{1,2\},\{3,4\},\{5,6\}$.