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Is the $0$-error capacity of $7$-cycle:

$(1)$ known to be of form $7^q$ for some $q\in \mathbb Q$?

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    $\begingroup$ I voted down because it does not show research effort. Simple Google search could answer your question: nothing is known about the Shannon capacity of C_7. $\endgroup$
    – Boris Bukh
    Commented Oct 16, 2014 at 13:49

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Indeed as Daniel pointed out, the Shannon capacity of $C_7$ is still unknown. Also, to say something closer to your question, its specific form is still unknown (in my knowledge).

Please see

T. Bohman, A limit theorem for the Shannon capacity of odd cycles II, Proc. Amer. Math. Soc. 133 (2005), no. 2, 537-543.

or the even more specific

A. Vesel, J. Zerovnik, Improved lower bounds on the Shannon capacity of C_7, Information Processing Letters, Volume 81, Issue 5, 16 March 2002, Pages 277–282.

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According to http://www.stahlke.org/dan/publications/phd-thesis.pdf (bottom of page 5) the Shannon capacity of the heptagon is still unknown.

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  • $\begingroup$ sorry my question was more about the form. $\endgroup$
    – Turbo
    Commented Oct 16, 2014 at 2:55

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