On The Relationship Between The Rank Of Mixed Graphs And Its Underlying Graphs

Spectral graph theory is an important research field of algebraic graph theory and combinatorial matrix theory.Hermitian spectral theory of mixed graph is a hot topic of spectral graph theory,mainly studies the relationships between the Hermitian spectral properties and the structural parameters of mixed graphs,discusses how the structures of graphs are characterized by their spectra.The Hermitian rank of mixed graph is an important spectral parameter of mixed graph.In this thesis,we mainly discuss the relationships between the Hermitian ranks of mixed graphs and its oriented properties.Mixed graph can be obtained from a simple graph by oriented a subset of its edge set.From a simple graph to a mixed graph by making an orientation,corresponding rank is changed.In this thesis,we study the relation between the Hermitian ranks of mixed graphs and the ranks of their underlying graphs,give the upper bound and lower bound of the Hermitian ranks of mixed graphs,and characterize the extremal graphs which can achieve the upper bound and lower bound,respectively.The thesis can be organized as follows.In the first Chapter,we firstly introduce the research background of graph theory and spectral graph theory,then we introduced some notations and concepts used in the later.Finally,we introduce the research problem,it developments and our main results.In the second Chapter,we give the upper bound of Hermitian ranks of mixed graphs and characterize the graphs which attain the upper bound.In the final chapter,we give the lower bound of Hermitian ranks of mixed graphs and characterize the graphs which attain the lower bound.