| The rank or nullity of a graph has always been the key research object of experts in graph spectra theory.Graph spectra theory studies the spectral properties of graphs by means of adjacency matrices,Laplacian matrices and unsigned Laplacian matrices.The rank of a graph is defined as the rank of its adjacency matrix,the nullity of a graph is defined as the nullity of its adjacency matrix(or the multiplicity of the zero eigenvalue of its adjacency matrix),the eigenvalue of a graph is defined as the eigenvalue of its adjacency matrix.In chemistry,a conjugated hydrocarbon can be represented by its molecular graph,where the vertices of represent the carbon atoms,and the edges of represent the carbon-carbon bonds of the conjugated hydrocarbon.For some kinds of molecules,the nullity equals zero is a necessary condition for their chemical stability.We have known that all connected graphs of rank 2 are complete bipartite graphs,and all graphs of rank 3 are complete tripartite graphs,Gerard J.Chang et al.characterized graphs with rank 4 and 5.However,there are few researches on the characterization of bicyclic graphs with a given rank.This thesis first introduces the research background and significance of graph theory,the research progress at home and abroad and the concepts and symbols involved in this thesis.Secondly,this thesis introduces the nullity of a kind of unicyclic graph with pendant path.According to the number of vertices contained in the cycle,it is divided into odd cycle graphs and even cycle graphs,then we will study the nullity separately.Then in this thesis,we characterize bicyclic graphs of rank 6 and rank 7with two triangles.According to zero,one or two common vertices in the two triangles,the bicyclic graphs are divided into three kinds for discussion.Finally,we summarize the main content of the thesis and put forward the research direction in the future.Figure: sixteen Reference: twenty-three... |
