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. Lov\'asz famously proved that the Shannon capacity of $C_5$ (the 5-cycle) is at most $\sqrt{5}$ via his theta function. This bound is achieved by a simple linear code over $\mathbb{F}_5$ mapping $x \mapsto 2x$. 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 $\sqrt{5}$. Our method applies more generally to Cayley graphs over the additive group of finite fields $\mathbb{F}_q$, giving an upper bound on the linear Shannon capacity. We compare this bound to the Lov\'asz 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.