Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: An Infinite Family of Graphs with the Same Ihara Zeta Function. | An Infinite Family of Graphs with the Same Ihara Zeta Function Christopher Storm Department of Mathematics and Computer Science Adelphi University cstorm@ Submitted Aug 17 2009 Accepted May 25 2010 Published Jun 7 2010 Mathematics Subject Classification 05C38 Abstract In 2009 Cooper presented an infinite family of pairs of graphs which were conjectured to have the same Ihara zeta function. We give a proof of this result by using generating functions to establish a one-to-one correspondence between cycles of the same length without backtracking or tails in the graphs Cooper proposed. Our method is flexible enough that we are able to generalize Cooper s graphs and we demonstrate additional families of pairs of graphs which share the same zeta function. 1. Introduction In 2009 Cooper described an infinite family of non-isomorphic pairs of graphs which she conjectured had the same Ihara zeta function 2 . In this note we provide a proof of Cooper s conjecture. We do so by using the definition of the Ihara zeta function directly as opposed to using determinant expressions for the zeta function. We will use bivariate generating functions to establish a one-to-one degree preserving correspondence between the sets used to build the Ihara zeta function. We refer the reader to 8 for a reference on generating functions. In the remainder of this section we introduce the Ihara zeta function define Cooper s graphs and state our main result. In Section 2 we develop the necessary tools and provide a proof of our main result. We conclude that section with some remarks on generalizing the family of graphs which have the same Ihara zeta function. A graph X V E is a finite nonempty set V of vertices and a finite multiset E of unordered pairs of vertices called edges. We allow edges of the form u u called loops. We also allow an edge u v to be repeated more than once as an element of E and refer to this as a multiple edge. the electronic journal of combinatorics 17 2010 R82 1
