The Fuzzy Cycle In Direct Product Of Fuzzy Graphs

In this thesis,we deal with the properties of the direct product of two fuzzy graphs,and present a detailed investigation into fuzzy cycles in the direct product.Firstly,we carry out research on the relations between fuzzy cycles in direct product and internal nodes of two fuzzy graphs involved,and give sufficient conditions for the direct product to have a fuzzy cycle.It is shown,with our investigation,that there exists a fuzzy cycle in the direct product of two fuzzy graphs both of which has an internal node,and that there is no fuzzy cycle while neither of them has internal node.Secondly,we study the direct products of some special types of fuzzy graph.More precisely,based on acyclic or cyclic fuzzy graphs,three classes of the direct products of fuzzy graphs attached,namely the direct product of two acyclic graphs,two cyclic fuzzy graphs,and a cyclic and an acyclic fuzzy graph.It is shown,as a result of our investigation,that every cycle in the direct product of two acyclic fuzzy graphs is a fuzzy cycle.As for cycle,there exits a fuzzy cycle in the direct product of two fuzzy graphs whose underlying graph are cycles,and every arc is on a certain fuzzy cycle.Moreover,there exists an even fuzzy cycle in the direct product of two even cycles,and there exists an odd fuzzy cycle in the direct product of two odd cycles.As for fuzzy cycle,the cycles in the direct product of two fuzzy cycles are still fuzzy cycles.In the case of acyclic fuzzy graphs,we show that every arc in the direct product of an acyclic fuzzy graph and an odd cycle is on a certain fuzzy cycle.While every cycle in the direct product of an acyclic fuzzy graph and a fuzzy cycle is a fuzzy cycle.