tailieunhanh - Báo cáo toán hoc:" Properties determined by the Ihara zeta function of a Graph "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài:Properties determined by the Ihara zeta function of a Graph. | Properties determined by the Ihara zeta function of a Graph Yaim Cooper Department of Mathematics Princeton NJ USA yaim@ Submitted Nov 1 2008 Accepted Jun 22 2009 Published Jul 9 2009 Mathematics Subject Classification 05C99 Abstract In this paper we show how to determine several properties of a finite graph G from its Ihara zeta function Zg u . If G is connected and has minimal degree at least 2 we show how to calculate the number of vertices of G. To do so we use a result of Bass and in the case that G is nonbipartite we give an elementary proof of Bass result. We further show how to determine whether G is regular and if so its regularity and spectrum. On the other hand we extend work of Czarneski to give several infinite families of pairs of non-isomorphic non-regular graphs with the same Ihara zeta function. These examples demonstrate that several properties of graphs including vertex and component numbers are not determined by the Ihara zeta function. We end with Hashimoto s edge matrix T. We show that any graph G with no isolated vertices can be recovered from its T matrix. Since graphs with the same Ihara zeta function are exactly those with isospectral T matrices this relates again to the question of what information about G can be recovered from its Ihara zeta function. 1 Introduction and Statement of Results In this paper we study properties of graphs which are determined by the Ihara zeta function. Here we allow multiple edges and loops in graphs. The Ihara zeta function defined below was introduced by Ihara in 1966 4 and in its present form associates to each finite graph a zeta function. In the remainder of this paper we will generally refer to this function simply as the zeta function. It has long been known that for any graph G whose minimal degree is at least 2 the number of edges of G can be computed from ZG u by considering the degree of the polynomial ZG u -1 1 . It is natural to Supported by NSF grant DMS-0353722 and Louisiana