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 $x \mapsto 2x$. Motivated by this, we introduce the notion of $\textit{linear Shannon capacity}$ 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 $\mathbb{F}_q$. We compare our bound to the Lov\'asz theta function, showing that they match for self-complementary Cayley graphs (such as $C_5$), and that our bound is smaller in some cases. We also exhibit a quadratic gap between linear and general Shannon capacity for some graphs.

中文翻译：

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凯莱图的线性香农容量

图的香农容量是零误差信息理论中的一个基本量，用于测量图幂中独立集的增长率。尽管得到了充分研究，但这个数量仍然存在几个谜团。Lov\'asz 通过他的 theta 函数证明了 $C_5$（5 周期）的香农容量至多为 $\sqrt{5}$。这个界限是通过 $\mathbb{F}_5$ 映射 $x \mapsto 2x$ 上的简单线性代码实现的。受此启发，我们引入了图的 $\textit{linear Shannon capacity}$ 的概念，这是将自身限制为线性代码时可实现的最大速率。我们基于多项式方法给出了一个简单的证明，即 $C_5$ 的线性香农容量为 $\sqrt{5}$。我们的方法更普遍地适用于有限域 $\mathbb{F}_q$ 的可加组上的 Cayley 图。我们将我们的界限与 Lov\'asz theta 函数进行比较，表明它们与自互补 Cayley 图（例如 $C_5$）匹配，并且在某些情况下我们的界限更小。我们还展示了一些图的线性和一般香农容量之间的二次差距。