N~?-rank Of Mixed Graphs With Small Dimensions

Spectral graph theory is an important research direction that both algebraic graph theory and combinatorial matrix theory pay attention to.It mainly studies the relationship between the spectrum of the graph and the structure of the graph.In the past few decades,the rank of a graph as an important spectral parameter has always attracted the attention of researchers.Mixed graphs are obtained by incomplete orientation of simple graphs.Since B.Mohar and Xueliang Li independently proposed the Hermite adjacency matrix of mixed graphs in 2015,the study of Hermite spectrum of mixed graphs has quickly become a new research hotspot in spectrogram theory.The Hermite spectrum theory of the mixed graph mainly establishes the relationship between the Hermite spectrum parameters of the mixed graph and the structural properties of the mixed graph.The spectral properties of the mixed graph are used to describe its structural properties,and the Hermite rank of the mixed graph is an important spectral parameter of the mixed graph.In 2020,B.Mohar defined a new Hermitian matrix N? for mixed graphs.In this paper,we discuss the N?-rank of mixed graphs,determining the N?-ranks of mixed trees,mixed cycles and mixed complete bipartite graphs.Moreover,all mixed graphs with N?-rank 2 and 3 and unicyclic graphs with rank 4 and 5 are characterized,respectively.The structure of this article is as follows:The first chapter mainly introduces the research background of the mixed graph N?-rank,some related concepts and symbols,as well as the basic problems,progress and results of the research in this paper.In the second chapter,the main research on the mixed graph The N?-rank of the graph describes all N?-connected mixed graphs with ranks 2 and 3.In Chapter 3,we mainly describe all N?-unicyclic graphs with ranks 4 and 5.