Ihara-zeta Function Of The Join Of Semi-regular Bipartite Graphs And Its Applications

The zeta function of a graph was first proposed by Ihara,he studied the zeta function of regular graphs by p-adic group.We also called it the Ihara-zeta function of a graph.Then a lot of results about the zeta function of a graph were obtained:Hashimoto discussed the zeta function of semi-regular bipartite graphs,Bass generalized the zeta function of regular graphs to general finite graphs and converted an infinite product to a polynomial related to the adjacency matrix and the degree matrix and so on,which was a very important result and provided a simple method for obtaining the zeta function of general finite graphs.The zeta function of a graph can determine some invariants of the graph,for example,Northshield found that the number of spanning trees in a general finite graph could be expressed as the derivative(at 1)of a determinant which is related to the zeta function of a graph.Based on Bass’ result,many scholars determined expressions of the Ihara-zeta functions of various families of graphs,such as the line graphs of semi-regular bipartite graphs,the middle graphs of semi-regular bipartite graphs,the cone of regular graphs,the r-join of regular graphs,the cone of semi-regular bipartite graphs and the corona graphs.This is a basic direction of the study on zeta function of a graph.In addition,whether the adjacency spectrum of a graph and the zeta function of a graph can be determined by each other is also a basic direction about the zeta function of a graph.For some special graphs,their zeta functions are the same if and only if they have the same adjacency spectrum.This article mainly consists of the following three results:(1)The Ihara-zeta function of the join of two semi-regular bipartite graphs is obtained by the techniques of matrix and determinant operations.According to the relation between the zeta function and the number of spanning trees of the join of semi-regular bipartite graphs,we obtain the number of spanning trees of the join of semi-regular bipartite graphs,and it is verified by another method.(2)The adjacency spectrum of the join of semi-regular bipartite graphs is obtained,and furthermore we prove that the adjacency spectrum of the join of semiregular bipartite graphs can determine the zeta function of the join of semi-regular bipartite graphs.(3)The convergence radius of the join of two simple semi-regular bipartite graphs is discussed.