Several Problems About The Nullity And Binary Rank Of Graphs

In the 1980s, based on of Gustafson's theoretical proof, Fiedler and Markham first formulated the nullity of graphs theory. Later Markham and Fiedler will abstract theory into research matrix, makes this theory in real matrix and complex matrix has applications. Since then, domestic and foreign scholars of the nullity of graphs to chart makes a deep research, and obtained a series of significant conclusions.The research for the nullity of graphs has a still olderchemical origin.Longuet-Higgins says a bipartite graph G (corresponding to a alternant hydrocarbons), if is singular, it means the molecule is unstable, this problem to the unbipartite graph (corresponding to the alternant hydrocarbons) also is significant. In addition, the research of binary rank of graphs have also made great development with the number of chromosomes, and the rank has relevant with it. The biggest vertice has been studied, this part VanNuffelen further demonstrated by the chromosome number of the graph,κ(G)≤rk2 (G),κ(G) is the chromosome number. At present about figure the nullity and binary rank,many scholars had in-depth research, but still remains the singularity problems and s the singular figure depicting problems still not getting good solution. In recent years, relevant chart of the nullity of graphs research is very active, especially figure for some special graphs and the graph with less edges, still not getting good solution.This paper focusing on the following main problems began research:1,the nullity of graphs, the research on around that the nullity of bicyclic graph with n-5, the nullity of subdivision graphs and the nullity of graphs with pendant trees.2,The binary rank of the adjacency matrix, figure the graphs with binary rank 2.As we know, the nullity of graphs is refers to the multiplicity of the eigenvalue zero in its spectrum,notes forη(G)> 0. Ifη(G)> 0, then A(G) is singular,otherwise is unsingular. For the nullity of graphs research has wider range, such as the research of trees,the unicyclic graphs, the bicyclic graphs and the linear graphs. For binary rank of the research is also has made progress, Godsil and Royle research provesκ(G)≤2r+1, including 2r is the graph binary rank.The thesis is organized as follows:the first chapter give this paper needed to prepare knowledge and research background, the second chapter gives the nullity of bicyclic graphs with n-5, the nullity of subdivision graphs and the nullity of graphs with pendant trees. The third chapter gives the background knowledge of binary rank and the graphs with binary rank 2.