Graph Theory And Topology, Graph Theory And Algebraic Cross-cutting Issues, Case Studies,

There are inflicational ways of thinking between different disciplines in mathematical science, which is an important tendency in the development of mathematics. Thus the dissertation, with this tendency as the center, focuses on the case studies of problems of overlap between Graph Theory and Topology, as well as Graph Theory and Algebra. In view of historical development, the present paper studies that the kind of overlapping fusion may cause the significant discoveries, even as emergence of inter-disciplines, using the succession analysis to each question and following the deduction rules. It has the special significance to expounded mathematics law of development.Concretely, the dissertation consists of four parts. How to surmount the measure idea, and how to face towards analysis situs are observed, during the fist part from Polyhedral Formula to Euler-Poincaré's number. In the second part, the evolution from Guthrie's question to Hadwiger's conjecture is involved, particularly analyzing how Wagner theorem initiated Hugo Hadwiger to propose the famous Conjecture. There inspectes the coloring question following its two development routes; On the one hand , curved surface showing topological characteristic is considered as a result of topological investigation, the chromatic number theory is being formed,which is a milestone in toplogical graph theory. It shows that, on the other hand, the introduction of Graph Minor has developed new visions for future research, thus the colouring question can obtain the further discrimination and generalization. The first two parts are the cases of problems of overlap between Graph Theory and Topology. Part III is based on the algebra of circuits, explaining how a fundamental set of circuits may be constructed and how to infuse algebra ideas into the study of graphs, and how to initiate Whitney present matroid (1935). That is a fine beginning for further studying. There introduces one theory which relates to the permutation group from counting trees, the simplest graphs, to Pólya's enumeration theorem in Part IV. The algebra of circuits and counting trees are two classical questions about the Graph Theory and Algebra. These four parts as a whole illustrate inflicational ways of thinking between different disciplines in mathematical science, it also inspects that, simultaneously, the historical perspective and the developing process of these overlapping questions manifests unification of mathematics. It is valuable to understand the tendency of ideological infiltration of modern mathematics.