Studies On Spectra And Zeta Functions Of Graphs

The spectral theory is one of the main research fields in the graph theory and the combinatorial matrix theory,which is widely used in quantum chemistry,physics,computer science and information science.The zeta functions of graphs can be considered as an analogue and an extension of zeta functions of a number theory.It is found that the zeta functions on graphs are very important to the spectra of graphs.This thesis mainly consists of the following three parts.1.In this part we mainly discuss some problems between the spectra and zeta functions of graphs.In general,the zeta function does not determine the adjacency spectrum of an arbitrary graph and vice versa.However,the zeta function and the adjacency spectrum can determine each other for special graph families.Based on the above,the zeta function of the cone over a semiregular bipartite graph is obtained,and we present the cones over semiregular bipartite graphs have the same adjacency spectrum if and only if they have the same Ihara zeta function.Moreover,the convergence of the zeta function of this family of graphs is considered.Second,we present the decomposition formulae for the zeta functions of families of several corona-type graphs,and we construct some families of graphs with the same zeta function by the adjacency spectra of graphs.Finally,we show that the complexity of a finite connected graph can be obtained from the partial derivatives at some points of a determinant in terms of the Bartholdi zeta function of .Moreover,the second order partial derivatives at these points of this determinant can all be expressed as the linear combination of Kirchhoff indices based on resistance distance of .2.The concept of hypergraph coverings over a hypergraph is introduced,and we generate all hypergraph coverings by permutation voltage assignments of the edge-colored graph or the incidence graph of a hypergraph.Consequently,we show two explicit decomposition formulae for the zeta function of any hypergraph covering which indicates that the zeta function of any hypergraph covering divides the zeta function of this hypergraph.3.First,by using the permutation voltage assignments of a graph and the symmetric group theory,the Laplacian characteristic polynomial and the normalized Laplacian characteristic polynomial of any graph covering over a graph are investigated.As applications,adopting the algebra method the Kirchhoff index,the multiplicative degree-Kirchhoff index and the complexity of any connected covering over a connected graph are derived.Next,we completely describe the normalized Laplacian eigenvalues and corresponding eigenvectors on the quadrilateral graph and th quadrilateral iterative graph.As applications,the significant formulae to calculate the multiplicative degree-Kirchhoff index,the Kemeny's constant and the number of spanning trees of the quadrilateral graph and the quadrilateral iterative graph are derived.