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arXiv 2103.06350math.CO
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网图的归纳

Inducibility of the Net Graph

Adam Blumenthal, Michael Phillips

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摘要：

一个图$F$称为分形子，如果对于所有的$n$，使$F$在$n$顶点上的诱导拷贝数最大化的图是$F$的平衡迭代爆破.虽然网络图是不是一个分形，我们表明，该网络几乎是一个分形。设$N(n)$是网图在$n$顶点上的导出副本的最大数目。对于足够大的$n$，我们表明，$N(n) = x_1\cdot x_2 \cdot x_3 \cdot x_4 \cdot x_5 \cdot x_6 + N(x_1) + N(x_2) + N(x_3) + N(x_4) + N(x_5) + N(x_6)$，其中$σx_i = n$和所有$x_i$是尽可能相等的。此外，我们证明了最大化$N(6^k)$的唯一图是平衡迭代爆破的网络$k$足够大。我们通过更仔细的计数和数值优化技术扩展了标准的标志代数和稳定性技术。

英文摘要：

A graph $F$ is called a fractalizer if for all $n$ the only graphs which maximize the number of induced copies of $F$ on $n$ vertices are the balanced iterated blow ups of $F$. While the net graph is not a fractalizer, we show that the net is nearly a fractalizer. Let $N(n)$ be the maximum number of induced copies of the net graph among all graphs on $n$ vertices. For sufficiently large $n$ we show that, $N(n) = x_1\cdot x_2 \cdot x_3 \cdot x_4 \cdot x_5 \cdot x_6 + N(x_1) + N(x_2) + N(x_3) + N(x_4) + N(x_5) + N(x_6)$ where $σx_i = n$ and all $x_i$ are as equal as possible. Furthermore, we show that the unique graph which maximizes $N(6^k)$ is the balanced iterated blow up of the net for $k$ sufficiently large. We expand on the standard flag algebra and stability techniques through more careful counting and numerical optimization techniques.

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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