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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Bartholdi Zeta Functions for Hypergraphs. | Bartholdi Zeta Functions for Hypergraphs Iwao SATO Oyama National College of Technology Oyama Tochigi 323-0806 JAPAN e-mail isato@oyama-ct.ac.jp Submitted Oct 22 2006 Accepted Dec 19 2006 Published Jan 3 2007 Mathematical Subject Classification 05C50 15A15 Abstract Recently Storm 8 defined the Ihara-Selberg zeta function of a hypergraph and gave two determinant expressions of it. We define the Bartholdi zeta function of a hypergraph and present a determinant expression of it. Furthermore we give a determinant expression for the Bartholdi zeta function of semiregular bipartite graph. As a corollary we obtain a decomposition formula for the Bartholdi zeta function of some regular 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 I uv 2 E G g. For e u v 2 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 D or G is a sequence P e1 en of n arcs such that e-i 2 D G t eị o ei 1 1 i n 1 . If e-i vi_1 Ui for i 1 n then we write P v0 v1 vn_1 vn . Set I P 1 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 eyt 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 cycle of a cycle C e1 en is the cycle C _1 en1 ef1 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 C is prime if it is not a multiple of a strictly smaller cycle. Note that each equivalence THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R00 1 class of prime reduced cycles of a .
