New lower bound on the Shannon capacity of C7 from circular graphs

TL;DR

This paper improves the lower bound on the Shannon capacity of the cycle graph C7 by constructing a large independent set in its fifth strong product power using computational methods and properties of circular graphs.

Contribution

It introduces a new independent set in the fifth power of C7, leading to a better lower bound on its Shannon capacity, utilizing computational techniques and circular graph properties.

Findings

Established an independent set of size 367 in the fifth power of C7.

Derived a lower bound on Shannon capacity: > 3.2578.

Used computational methods and properties of circular graphs for the construction.

Abstract

We give an independent set of size $367$ in the fifth strong product power of $C_7$, where $C_7$ is the cycle on $7$ vertices. This leads to an improved lower bound on the Shannon capacity of $C_7$: $\Theta(C_7)\geq 367^{1/5} > 3.2578$. The independent set is found by computer, using the fact that the set $\{t \cdot (1,7,7^2,7^3,7^4) \,\, | \,\, t \in \mathbb{Z}_{382}\} \subseteq \mathbb{Z}_{382}^5$ is independent in the fifth strong product power of the circular graph $C_{108,382}$. Here the circular graph $C_{k,n}$ is the graph with vertex set $\mathbb{Z}_{n}$, the cyclic group of order $n$, in which two distinct vertices are adjacent if and only if their distance (mod $n$) is strictly less than $k$.