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arXiv 1808.07438math.COcs.ITmath.IT
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循环图中C7的Shannon容量的新下界

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

Sven Polak, Alexander Schrijver

备注：
5 pages. Some changes have been made based on comments of the referees. Accepted for publication in Information Processing Letters
发表信息：
Information Processing Letters, 143 (2019), 37-40
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摘要：

给出了$C_7$的五次强积幂中的一个独立集，其大小为$367$，其中$C_7$是$7$顶点上的圈.这导致$C_7$：$Θ(C_7)\geq 367^{1/5} > 3.2578$的香农容量的改进的下界。利用循环图$C_{108,382}$的五次强积幂中集合$\{t \cdot (1,7,7^2,7^3,7^4) \,\, | \,\, t \in \mathbb{Z}_{382}\} \subseteq \mathbb{Z}_{382}^5$是独立的这一事实，用计算机求出了独立集.循环图$C_{k,n}$是顶点集为$\mathbb{Z}_{n}$的图，$n$阶循环群，其中两个不同的顶点相邻当且仅当它们的距离（mod $n$）严格小于$k$。

英文摘要：

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 $C_7$: $Θ(C_7)\geq 367^{1/5} > 3.2578$. 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 $k$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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