tailieunhanh - Báo cáo toán học: "Bartholdi Zeta Functions of Fractal Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Bartholdi Zeta Functions of Fractal Graphs. | Bartholdi Zeta Functions of Fractal Graphs Iwao Sato Oyama National College of Technology Oyama Tochigi 323-0806 Japan e-mail isato@ Submitted Aug 12 2008 Accepted Feb 18 2009 Published Feb 27 2009 Mathematical Subject Classification 05C50 05C25 05C10 15A15 Abstract Recently Guido Isola and Lapidus 11 defined the Ihara zeta function of a fractal graph and gave a determinant expression of it. We define the Bartholdi zeta function of a fractal graph and present its determinant expression. 1 Introduction Zeta functions of graphs started from p-adic Selberg zeta functions of discrete groups by Ihara 14 . At the beginning Serre 20 pointed out that the Ihara zeta function is the zeta function of a regular graph. In 14 Ihara showed that their reciprocals are explicit polynomials. A zeta function of a regular graph G associated to a unitary representation of the fundamental group of G was developed by Sunada 22 23 . Hashimoto 13 treated multivariable zeta functions of bipartite graphs. Bass 3 generalized Ihara s result on zeta functions of regular graphs to irregular graphs. Various proofs of Bass theorem were given by Stark and Terras 21 Kotani and Sunada 15 and Foata and Zeilberger 5 . Bartholdi 2 extended a result by Grigorchuk 7 relating cogrowth and spectral radius of random walks and gave an explicit formula determining the number of bumps on paths in a graph. Furthermore he presented the circuit series of the free products and the direct products of graphs and obtained a generalized form Bartholdi zeta function of the Ihara -Selberg zeta function. All graphs in this paper are assumed to be simple. Let G be a connected graph with vertex set V G and edge set E G and let R G u v v u I uv 2 E G g be the set of oriented edges or arcs u v v u directed oppositely for each edge uv of G. For e u v 2 R G u o e and v t e are called the origin and the terminal of e respectively. Furthermore let e-1 v u be the inverse of e u v . A path P of length n in G is a .