Finite Graph Zeta Function

We consider the element discussion of Ihara-type zeta function and their generalization for finite,possibly irregular graphs.Given a finite graph,there is a group at every vertex.The group may be a finite group(order＞1) not necessarily be an unitary group.Then the definition of loop has been changed. In this paper,we will adapt the definition of loop of(G,X) and calculate the number of non-backtracking,tailless loops.Then we will give out two definitions:The first is more simply,we define an equivalent class over the above loop, eg.two cycles C=ay₁y₂y₃ and D=y₁y₂y₃a,we say C～D.In this case,zeta function is a real function.The second definition is a directed generalization of Ihara zeta functions.In other words,Ihara zeta functions is its special case.For the more,we will give some explicit examples of zeta functions for finite graph.