Study On Singular Graphs And Nonsingular Graphs

Algebraic graph theory is an important branch of combinatorial mathematics,and singularity is a hot research problem in graph theory.The rank of the adjacency matrix A(G)is closely related to singularity,and the rank of the graph G is the rank of the adjacency matrix A(G).If the rank of the adjacency matrix A(G)is less than the order of the graph G,then the graph G is called singular graph.If the rank of the adjacency matrix A(G)is equal to the order of the graph G,then the graph G is said to be nonsingular.In this paper,it is defined that a connected graph G of order n(≥ 2)is perfectly nonsingular if all connected induced subgraphs of order k(≥ 2)are nonsingular.Connected graphs of order n(≥ 3)are perfectly singular if all connected induced subgraphs of order k(≠2)are singular.The connected graph G is perfectly staggered when all even order connected induced subgraphs of a connected graph G are not singular graphs,but all odd order connected induced subgraphs of a connected graph G are singular graphs.In this paper,we describe perfect nonsingular graphs,perfect singular graphs and perfect staggered graphs according to the singularity of cycle Cn,path Pn,complete graphs Kn and complete bipartite graphs Km,n-m.The thesis is divided into six chapters:The first chapter firstly introduces the development history of graph theory,then introduces the background and significance of graph theory and adjacency spectrum,and finally introduces the problems and the research progress of singularity in this paper.The second chapter firstly introduces some known basic concepts in graph theory,and makes symbolic explanation,and then introduces the basic concepts of the new definition graph.Finally,through the understanding of the basic knowledge of algebra and graph theory,this chapter introduces some special graphs related to the singularity of the basic lemma.Chapter three and Chapter four firstly prove the equivalence between perfect nonsingular graphs(perfect singular graphs)and weak perfect nonsingular graphs(weak perfect singular graphs)according to the basic lemma,and describe them,and then give the inference.Finally,the perfect nonsingular graph and perfect singular graph are applied and summarized respectively.The fifth chapter mainly limits the singularity and nonsingularity,and uses the basic lemma to judge the relation between perfect and weak perfect staggered graphs.Finally,the perfect staggered graph is applied and summarized.The sixth chapter summarizes the whole text and points out the innovation and deficiency of this article.Figure[21]Reference[81]...