Determine the Shannon capacity of the seven-cycle

Determine the exact Shannon capacity \(\Theta(C_7)\) of the seven-cycle, which is currently known only to satisfy \(3.2578<\Theta(C_7)<3.3177\).

Background

The paper notes that Shannon capacity is difficult to calculate or approximate for general graphs. In particular, the exact value for the seven-cycle C7C_7 remains unresolved, despite the graph's small size and simple structure. The stated bounds are provided by the Lovász theta function and represent the best known bounds at the time of writing.

References

Indeed, the Shannon capacity of even some small and simple graphs is still unknown, as is the case of e.g. the $7$-cycle $C_7$ whose Shannon capacity is, at the time of writing, only known to lie within the following bounds $3.2578 < \Theta(C_7) < 3.3177$ .

— Characterizing the Lovasz theta function via walk generating functions  (2501.15277 - Wolff, 25 Jan 2025) in Section 1, Introduction