On The Number Of Hermitian Cospectral Classes Of Mixed Graphs

Spectral graph theory is an important research field of algebraic graph theory and combinatorial matrix theory.Hermitian spectral theory of mixed graph mainly studies the relationship between the Hermitian spectral property and the structural property of mixed graphs,discusses how the structural property of a graph is characterized by its spectral property.In recent years,the Hermitian spectral theory has received more and more attentions.In order to achieve this goal,the first question is to study how much of a mixed graph can be reflect by its Hermitian spectrum,that is,to study the mixed graphs determined by their Hermitian spectra.For the mixed graphs those can not be determined by their Hermitian spectra,determining the num-ber of cospectral classes is also a natural and important problem.In this thesis,we determine the number of cospectral classes of bicyclic mixed graphs with the same underlying graph,classifing bicyclic graphs in terms of the number of cospectral classes,and give an upper bound of the number of cospectral classes for mixed graphs with a given planar ?-cyclic graph as underlying graph.The thesis can be organized as follows.In Chapter 1,we firstly introduce the background of graph theory and spectral graph theory,then introduce some concepts and notations,finally introduce the research problems,their developments and main results we get in this thesis.In Chapter 2,we firstly introduce some preliminaries which will be used in the later,then determine the number of cospectral classes of bicyclic mixed graphs with the same underlying graph,classifing bicyclic graphs in terms of the number of cospectral classes.In Chapter 3,we firstly introduce some preliminaries,then give an upper bound of the number of cospectral classes for mixed graphs with a given planar ?-cyclic graph as underlying graph,and construct a class of planar ?-cyclic mixed graphs that can achieve the upper bound.