On the Shannon capacity of triangular graphs (Q1953512)

Summary: The Shannon capacity of a graph \(G\) is \(c(G)=\sup_{d\geq 1}(\alpha(G^d))^{\frac{1}{d}},\) where \(\alpha(G)\) is the independence number of \(G\). The Shannon capacity of the Kneser graph \(\mathrm{KG}_{n,r}\) was determined by Lovász in 1979, but little is known about the Shannon capacity of the complement of that graph when \(r\) does not divide \(n\). The complement of the Kneser graph, \(\overline{\mathrm{KG}}_{n,2}\), is also called the triangular graph \(T_n\). The graph \(T_n\) has the \(n\)-cycle \(C_n\) as an induced subgraph, whereby \(c(T_n) \geq c(C_n)\), and these two families of graphs are closely related in the current context as both can be considered via geometric packings of the discrete \(d\)-dimensional torus of width \(n\) using two types of \(d\)-dimensional cubes of width \(2\). Bounds on \(c(T_n)\) obtained in this work include \(c(T_7) \geq \root 3 \of {35} \approx 3.271\), \(c(T_{13}) \geq \root 3 \of {248} \approx 6.283\), \(c(T_{15}) \geq \root 4 \of {2802} \approx 7.276\), and \(c(T_{21}) \geq \root 4 \of {11441} \approx 10.342\).