Several Issues On Interval-valued Fuzzy Graphs And Extremes Vague Figure

Fuzzy graph is a generalization of classical graph, so far fuzzy graph theory has been applied in clustering analysis, systems analysis, transportation, database theory, network analysis, information theory, economics, etc. As two significant generalizations of fuzzy graph, interval-valued fuzzy graph and bipolar fuzzy graph are actually believed to give more precision, flexibility and compatibility to the correlate system as compared to the classical and fuzzy graph models. This paper focuses on some problems about interval-valued fuzzy graph and bipolar fuzzy graph by the thought and method of lattice topology, such as operations aspect, categorical properties, individual sets and strong individual sets in nonstandard analysis. The main contents are as follows:Chapter1is a introduction. Some basic concepts and conclusions in interval-valued fuzzy graph, bipolar fuzzy graph as well as categorical theory are presented.Chapter2mainly introduces the operations aspect of interval-valued fuzzy graph. Firstly a correction is given to Akram and Dudck’s paper [57]. Next, four kinds of new product operations (called direct product, semi-strong product, strong product and lexi-cographic product) of interval-valued fuzzy graphs are defined, and rationality of these no-tions and some defined important notions on interval-valued fuzzy graphs, such as interval-valued fuzzy graph, interval-valued fuzzy complete graph, Cartesian product of interval-valued fuzzy graphs, composition of interval-valued fuzzy graphs, union of interval-valued fuzzy graphs, and join of interval-valued fuzzy graphs, are demonstrated by characteriz-ing these notions by their level counterparts graphs. it is proved that the class of strong interval-valued fuzzy graphs is closed under operations of cartesian product and composi-tion, but not under union and join. The conditions under which the class is closed under operations of union and join are obtained, connections between operations of composition, union, and join and operation of complement are also studied. Finally, the necessary and sufficient conditions of decomposing interval-valued fuzzy graph under operations carte-sian product and composition are obtained, and it is proved that interval-valued fuzzy graph can be decomposed under operation union and strong interval-valued fuzzy graph can be decomposed under operation join. The necessary and sufficient conditions of de-composing fuzzy graph under operations direct product, strong product and lexicographic product are also obtained. Chapter3focuses on categorical aspects of interval-valued fuzzy graph. Firstly, five categories IVFG, ReIVFG, SyIVFG, TrIVFG and FiIVFG are defined, and the relationships between them are studied. IVFG, ReIVFG, SyIVFG and TrIVFG are all proved to be topological categories on Set, FiIVFG is proved to be topological category on FiniSet. Structures of final and initial lift of the five categories are obtained. The formations of product and coproduct in the five categories and the formations of equalizer and coequalizer in IVFG are given. Next, interval-valued fuzzy graph-cotower, finite interval-valued fuzzy graph-cotower, reflective interval-valued fuzzy graph-cotower, symmetric interval-valued fuzzy graph-cotower and transitive interval-valued fuzzy graph-cotower are defined, and IVFGC, ReIVFGC, SyIVFGC, TrIVFGC and FiIVFGC are also defined. A one-to-one correspondence between interval-valued fuzzy graph (resp., reflective interval-valued fuzzy graph, symmetric interval-valued fuzzy graph, transitive interval-valued fuzzy graph, finite interval-valued fuzzy graph) and interval-valued fuzzy graph-cotower (rasp., reflective interval-valued fuzzy graph-cotower, symmetric interval-valued fuzzy graph-cotower, transitive interval-valued fuzzy graph-cotower, finite interval-valued fuzzy graph-cotower) is given, and category IVFG (resp., ReIVFG, SyIVFG, TrIVFG, FiIVFG) and IVFGC (resp., ReIVFGC, SyIVFGC, TrIVFGC, FiIVFGC) is isomorphic.Chapter4is devoted to bipolar fuzzy graph. Firstly, three kinds of new product operations (called direct product, semi-strong product and strong product) of bipolar fuzzy graphs are defined, and it is proved that the the direct product, semi-strong product or strong product of two strong bipolar fuzzy graphs is strong, but the converse conclusion does not hold, a counterexample is given. Next, a new definition of complement of bipolar fuzzy graph is obtained, this new definition is different to the old definition, it has two advantages which the old definition does not have:G is a bipolar fuzzy graph, then (?)=G, where (?) is the complement of G; Automorphism groups of G and G are identical. Properties of the new complement of bipolar fuzzy graph are presented.Chapter5For convenience of the application of nonstandard analysis and category theory in lattice-valued mathematics (including interval-valued fuzzy graph and bipolar fuzzy graph) and supernetwork theory, the categorical aspect of individual sets and strong individual sets which are basic definitions in nonstandard analysis are studied. The cate-gory of individual sets and the category of strong individual sets are proved to be much similar to the category of sets by using method of category theory. For example, for each cardinality a, there exists a individual set (resp., a strong individual set Xa) such that|Xα|=a, where|Xα|is the cardinality of Xα; Individual sets and strong individual sets are closed under the operations of subset and power; Both the category of nonempty in-dividual sets and the category of nonempty strong individual sets are complete monoidal topoi. Superstructure functor V and ultrapower functor HF are constructed and the fol-lowing conclusions are obtained:(1) For any nonempty strong individual sets X and Y g:Xâ†'Y is an injection (resp., a surjection) if and only if V(g) is an injection (resp., a surjection);(2) For any sets X and Y, g:Xâ†'Y is an injection (resp., a surjection) if and only if HF(g) is an injection (resp., a surjection).