Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A new determinant expression of the zeta function for a hypergraph. | A new determinant expression of the zeta function for a hypergraph Iwao Sato Oyama National College of Technology Oyama Tochigi 323-0806 Japan isato@ Submitted Mar 1 2009 Accepted Oct 19 2009 Published Oct 31 2009 Mathematical Subject Classification 05C50 15A15 Abstract Recently Storm 10 defined the Ihara-Selberg zeta function of a hypergraph and gave two determinant expressions of it by the Perron-Frobenius operator of a digraph and a deformation of the usual Laplacian of a graph. We present a new determinant expression for the Ihara-Selberg zeta function of a hypergraph and give a linear algebraic proof of Storm s Theorem. Furthermore we generalize these results to the Bartholdi zeta function of a hypergraph. 1 Introduction Graphs and digraphs treated here are finite. Let G be a connected graph and D the symmetric digraph corresponding to G. Set D G u v v u uv G E G . For e u v G D G set u o e and v t e . Furthermore let e-1 v u be the inverse of e u v . A path P of length n in G is a sequence P e1 en of n arcs such that ei G D G t ei o ei 1 1 i n 1 . If ei vi-1 vi for i 1 n then we write P v0 v1 vn-1 vn . Set P n o P o e1 and t P t en . Also P is called an o P t P -path. We say that a path P e1 en has a backtracking or a bump at t ei if e-pt ei 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 path of a path P e1 en is the path P-1 e-1 e-1 . We introduce an equivalence relation between cycles. Two cycles C1 e1 em and C2 f1 fm are called equivalent if fj ej k for all j. The inverse cycle of C is 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 multiple of B. A cycle C is reduced if both C and C2 have no backtracking. Furthermore a cycle THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R132 1 C is prime if it is not a multiple of a strictly smaller cycle. Note that each equivalence class of prime
