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arXiv 1602.02797math.COmath.GT
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生成树与Mahler测度

Spanning Trees and Mahler Measure

Daniel S. Silver, Susan G. Williams

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12 pages, 1 figure
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摘要：

有限连通图的复杂度是它的生成树的数目;对于非连通图，它是它的连通分支的复杂度的乘积。若$G$是一个具有余有限自由${\mathbb Z}^d$ -对称性的无限图，则其Laplacian多项式$Δ$的对数Mahler测度$m(Δ)$是$G$的有限项复杂性的指数增长率.它以$m(Δ({\mathbb G}_d))$为界，其中${\mathbb G}_d$是维度$d$的网格图。当$d$趋于无穷大时，增长率$m(Δ({\mathbb G}_d))$渐近于$\log 2d$。如果为$m(Δ(G))\ne 0$，则为$m(Δ(G)) \ge \log 2$。给出了平面图交错链环族的行列式增长率的一个应用。

英文摘要：

The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If $G$ is an infinite graph with cofinite free ${\mathbb Z}^d$-symmetry, then the logarithmic Mahler measure $m(Δ)$ of its Laplacian polynomial $Δ$ is the exponential growth rate of the complexity of finite quotients of $G$. It is bounded below by $m(Δ({\mathbb G}_d))$, where ${\mathbb G}_d$ is the grid graph of dimension $d$. The growth rates $m(Δ({\mathbb G}_d))$ are asymptotic to $\log 2d$ as $d$ tends to infinity. If $m(Δ(G))\ne 0$, then $m(Δ(G)) \ge \log 2$. An application to determinant growth rates of families of alternating links arising from planar graphs is given.

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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