| Fuzzy mathematics is a new subject and it is about the theories andmethods that relate to describe and deal with the problems of the fuzziness. Theapplication of fuzzy mathematics is very extensive and some graphs researchers alsocombined the fuzzy mathematics with the knowledge of graphs and got the concept offuzzy graphs. The theory of fuzzy graphs incurred quickly strong interest of thegraphs researchers and in recent years, the graphs researchers study constantly fuzzygraphs, making the theory of fuzzy graphs has been able to some with the theory ofgraph basis, graph energy and graph algebra, as a branch of graph theories. What ismore, the theory of fuzzy graphs is currently very active in the research of graphtheory. With the development of the theory of the fuzzy set, there was a problemwhich is the theory of the parameter tool is inadequate in the research. In order tosolve this problem, in1999, Molodtsov gave the definition of the soft set and thetheory of the soft set is used to deal with the uncertain information. In2001,Maji P K introduced the concept of the fuzzy soft set, because of this, the soft setresearchers had a broader space. In recent years, more and more theories related to thefuzzy soft set have been proposed, which making the theory of the fuzzy soft set hasbecome an active new topic in the current. This paper introduces the theory of fuzzysoft graphs in the basis of fuzzy graphs and the fuzzy soft set. Applying the theory ofthe fuzzy soft set to the theory of fuzzy graphs has opened up a new direction for theresearch of fuzzy graphs and also provided a new mathematical tool and a new usefulmethod, which is very important to the future study of fuzzy graph.In this paper, we discuss the definition and the operations of fuzzy soft graphs.In the first part of the paper, we give the definitions of fuzzy graphs and theiroperations and fuzzy soft set. In the second part of the paper, we give the definition offuzzy soft graphs, then, we discuss the definition and the properties of theconnectivity of fuzzy soft graphs. In the third part of the paper, we give the definitionsand the properties of the bargain, combine, sum, strong product, direct product orlexicographic product of fuzzy soft graphs and so on. In the four part of the paper, wegive the definition of M-strong fuzzy graphs and its properties. At last, we give theweakness of the paper and we had like someone to give a better answer to fuzzy softgraphs. |
