Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The Scattering Matrix of a Graph. | The Scattering Matrix of a Graph Hirobumi Mizuno Iond University Tokyo Japan Iwao Sato Oyama National College of Technology Oyama Tochigi 323-0806 Japan isato@ Submitted May 25 2008 Accepted Jul 16 2008 Published Jul 28 2008 Mathematics Subject Classification 05C50 15A15 Abstract Recently Smilansky expressed the determinant of the bond scattering matrix of a graph by means of the determinant of its Laplacian. We present another proof for this Smilansky s formula by using some weighted zeta function of a graph. Furthermore we reprove a weighted version of Smilansky s formula by Bass method used in the determinant expression for the Ihara zeta function of a graph. 1 Introduction Graphs treated here are finite. Let G V G E G be a connected graph possibly multiple edges and loops with the set V G of vertices and the set E G of unoriented edges uv joining two vertices u and v. For uv 2 E G an arc u v is the oriented edge from u to v. Set R G u v v u I uv 2 E G g. For b u v 2 R G set u o b and v t b . Furthermore let b v u be the inverse of b u v . A path P of length n in G is a sequence P b1 bn of n arcs such that b 2 R G t bi o bi 1 1 i n 1 where indices are treated mod n. Set I P 1 n o P o b1 and t P t bn . Also P is called an o P t P -path. We say that a path P b1 bn has a backtracking or back-scatter if bi 1 bi for some i 1 i n 1 . A v w -path is called a v-cycle or v-closed path if v w. The inverse cycle of a cycle C b1 bn is the cycle C bn b1 . We introduce an equivalence relation between cycles. Two cycles C1 e1 em and C2 f1 fm are called equivalent if there exists k such that fj Cj k for all j. The inverse cycle of C is in general not equivalent to C. Let C be the equivalence class which contains a cycle C. Let Br be the cycle obtained by going r times around a cycle B. Such a cycle is called a power of B. A cycle C is reduced if C has no backtracking. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R96 1 Furthermore a cycle C is primitive if it is
