分形图的一条迹线与Ihara zeta函数
A trace on fractal graphs and the Ihara zeta function
摘要：
从1968年伊原的工作，有越来越多的兴趣研究zeta函数的有限图，由须田，桥本，巴斯，斯塔克和特拉斯，水野和佐藤，仅举几个作者。然后，克莱尔和Mokhtari-Sharghi研究了zeta函数的无限图作用于一个离散的自同构群。所有这些处理的主要公式建立了zeta函数（最初定义为无穷乘积）和图的拉普拉斯算子之间的联系。在这篇文章中，我们考虑另一类无限图。它们是分形图，即它们具有自相似性。我们定义了这些图的zeta函数，并使用算子代数的机器，我们证明了一个行列式公式，它涉及zeta函数与Laplacian的图。我们还证明了函数方程，和一个公式，它允许zeta函数的zeta函数的有限子图的近似。
英文摘要：
Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a self-similarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs.
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