| Recently, fuzzy graph theory has been a generalization of classical graph, and hasnumerous applications in modern science and engineering. Since interval-value fuzzy set theoryis an extension of fuzzy set theory where [0,1]-valued membership degree is replaced byinterval-value in [0,1]that approximate the membership degree; then specific types ofinterval-valued fuzzy graphs have been introduced and investigated. Then interval-valued fuzzygraph is a generalization of fuzzy graph, are more precision and flexibility to solve and apply tosome uncertain problems as compared to the classical graph theory and fuzzy graph theory.In this paper, on the basis of fuzzy graph theory, as a starting point of having thediscussion of interval-valued fuzzy graph, discussed the properties of the edge, operations,complement graph, line graph and the control relation in this class of graphs. For establishingthe system of interval-valued fuzzy graph theory, provides many necessary references andtheoretical basis.Firstly,this paper simply introduce the concept of interval-valued fuzzy graph, and theconcepts of its complement graph, line graph and the degree of neighboring point and somebasic theories.Then, this paper primarily discusses the properties of edges of interval-valued fuzzygraphs, its simply operations, its complement graph, line graph and some simply controlrelations between interval-valued fuzzy graphs.With the problem about studying the properties of edges of interval-valued fuzzy graphs, isprimarily based on the connection of fuzzy graph theory, raise the thinking of investigationabout some special concepts and theories of edge, this paper gives one transform relationbetween the interval-value and value in [0,1]by means of introducing the order methods ininterval-values, immediately,“width” in this paper. So through the membership degree of edgein interval-valued fuzzy graph is replaced by “width”, discusses the connection betweeninterval-valued fuzzy graphs, then gives “the width of road”,“the strength of connection”,“cutedge” and so on.In this part of the interval-valued fuzzy graph’s simply operations, knows from the sum、and、bargain from the paper [12], this paper gives co-direct product, weak direct product, newdirect product. With this operation, also gives the properties and significance of the operationsin the interval-valued fuzzy graphs. This new operation way reflects a new relation between interval-valued fuzzy graphs, and with future investigates, it will develop to specific use.Then, under the concepts of complement graph, line graph and the degree of neighboringpoint in this paper, combines some studying thinkings in the theories, gives simply theories inthe third section and fourth section in the third chapter. In the last part of the third chapter, thispaper bases on the control theory, combines with the characteristics of interval-valued fuzzygraph, preliminary discussed the control relationship between interval-valued fuzzy graphs, andcombines with classic graph, by limiting the some conditions of interval-valued fuzzy graph,puts forward the interval-valued fuzzy bipartite graph; Then, discusses the interval-valued fuzzygraph’s control set and independent set.Finally, this paper summarizes the this paper mainly research content and its necessity, andthe combination of fuzzy graph theory research in information theory, the approximatecalculation, network theory and a variety of applications and its importance,and thedevelopment of this paper in the future. |
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