DHS Of Mixed Graphs

Spectral graph theory mainly studies the relationships between the struc-tural properties and the spectral properties of graphs,and characterizes the structure of a graph by its spectrum.One basic problem in spectral graph theory is DS problem,that is,discussing how a graph is determined by its spectrum.The graph containing undirected edges and directed edges is called the mixed graph,which can be obtained from simple graph by oriented a subset of its edge set.Mixed graphs can be considered as natural generalizations of simple graphs and oriented graphs.In 2015,Xueliang Li et al.and Mohar et al.independently introduced Hermitian adjacency matrix of mixed graphs,respectively.Then,this topic has received much attention.In this thesis,we mainly discuss some related problems of DHS,and characterize all mixed graphs with H-rank 3.Besides,we prove that all connected mixed graphs with H-rank 3 can be determined by their H-spectrum.This thesis is organized as follows.In chapter one,we firstly introduce research background of the spectral graph theory,some concepts and notations that will be used.Finally,we present the research problems,their develop-ments and the main results obtained in this thesis.In the next chapter,we discuss characteristic polynomials of mixed graphs and obtain some pertur-bation properties of characteristic polynomials of mixed graphs.In the last chapter,we discuss H-rank of mixed graphs and prove that all connected mixed graphs with H-rank 3 are DHS.Meanwhile,we construct infinitely many mixed graphs with H-rank 3 which are not DHS.