Linear Shannon Capacity of Cayley Graphs

The Shannon capacity of a graph is a fundamental quantity in zero-error information theory measuring the rate of growth of independent sets in graph powers. Despite being well-studied, this quantity continues to hold several mysteries. Lovasz famously proved that the Shannon capacity of C-5 (the 5-cycle) is at most root 5 via his theta function. This bound is achieved by a simple linear code over F-5 mapping x -> 2x. This motivates the notion of linear Shannon capacity of graphs, which is the largest rate achievable when restricting oneself to linear codes. We give a simple proof based on the polynomial method that the linear Shannon capacity of C-5 is root 5 Our method applies more generally to Cayley graphs over the additive group of finite fields F-q, giving an upper bound on the linear Shannon capacity. We compare this bound to the Lovasz theta function, showing that they match for self-complementary Cayley graphs (such as C-5), and that the bound is smaller in some cases. We also exhibit a quadratic gap between linear and general Shannon capacity for some graphs.