小世界Farey图中生成树的计数
Counting spanning trees in a small-world Farey graph
摘要：
生成树的问题与统计物理领域中的各种有趣的问题密切相关，但是确定一般网络中的生成树的数量在计算上是困难的。本文研究了一个顶点度服从指数分布的小世界网络中生成树的计数问题，该网络被称为Farey图，因为它与著名的Farey序列相关联。根据特殊的网络结构，给出了Farey图及其子图的Laplacian特征多项式的递推关系。然后，利用这里得到的这些关系，我们得到的确切数目的生成树的Farey图，以及近似的数值解的渐近增长常数表征的网络。最后，我们将我们的结果与以前研究的不同类型的网络进行了比较。
英文摘要：
The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform a study on the enumeration of spanning trees in a specific small-world network with an exponential distribution of vertex degrees, which is called a Farey graph since it is associated with the famous Farey sequence. According to the particular network structure, we provide some recursive relations governing the Laplacian characteristic polynomials of a Farey graph and its subgraphs. Then, making use of these relations obtained here, we derive the exact number of spanning trees in the Farey graph, as well as an approximate numerical solution for the asymptotic growth constant characterizing the network. Finally, we compare our results with those of different types of networks previously investigated.
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。
