Research On The Operations And Properties Of Graph Lattices And Fuzzy Graphs

A fuzzy graph is a fuzzification of a graph, which can also be looked as a weighted graph. This dissertation focuses on the studying of operations and properties of interval-valued fuzzy graphs, fuzzy soft graphs and graph lattices. Specific content is as follows:Chapter1introduces the basic concepts of the fuzzy graph, interval number, fuzzy soft set and those of lattice theory which will be used in the follwing chapters.Chapter2defines the concepts of cut point, cut edge and block of fuzzy graphs, and discusses the related properties, including the sufficient and necessary conditions for a point (resp., an edge) to be a cut point (resp., a cut edge) of a fuzzy graph, the sufficient and necessary conditions for a fuzzy graph with an order at least three (resp., a fuzzy circle) to be a block, and the conditions that the distance between vertexes can be a metric.Chapter3studies the operations and related properties of fuzzy soft graphs. Firstly, the concepts of fuzzy soft graph and connectivity of fuzzy soft graphs are given, and then properties of the connected fuzzy soft graphs are discussed. Secondly, we define some operations of fuzzy soft graphs, such as meet, join, complement, and, or, Cartesian product, strong product, direct product, lexicographic product, etc. Moreover, the related properties of these operations and the concept of cut operation in the fuzzy soft graphs as well as its properties are studied. Finally, similar to the fuzzy soft graphs, the definition of M-strong fuzzy soft graphs is given, and their connectivity and properties of operation are discussed.Chapter4mainly summarizes and proves the related properties of interval-valued fuzzy graphs. These include the properties of basic operations, the related properties of their complementary graphs, the properties of their line graphs etc. The control relationship of interval-valued fuzzy graphs has also been studied in this chapter.Chapter5gives the definition of graph lattice which is defined via the partial order relationship in the subgraph sets. Further, the idea of graph lattice is put forward to solve a guess on Ramsey problem in combinatorial mathematics.At last, the further work is given.