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. Lovász 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 → 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 Lovász 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.