| Paper ID | D3-S1-T1.1 |
| Paper Title |
Linear Shannon Capacity of Cayley Graphs |
| Authors |
Venkatesan Guruswami, Andrii Riazanov, Carnegie Mellon University, United States |
| Session |
D3-S1-T1: Combinatorics & Information Theory |
| Chaired Session: |
Wednesday, 14 July, 22:00 - 22:20 |
| Engagement Session: |
Wednesday, 14 July, 22:20 - 22:40 |
| Abstract |
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 $\F_5$ mapping $x \mapsto 2x$. This motivates the notion of \emph{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~$\F_q$, 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.
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