On total regularity of the join of two interval valued fuzzy graphs

uzzy graph theory is finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. In the applied field, the success of the use of fuzzy set theory depends on the choice of the membership function that we make. However, there are applications in which experts do not have precise knowledge of the function that should be taken. In these cases, it is appropriate to represent the membership degree of each element of the fuzzy set by means of an interval. From these considerations arises the extension of fuzzy sets called the theory of interval valued fuzzy sets. That is, fuzzy sets such that the membership degree of each element of the fuzzy set is given by a closed subinterval of the interval [0,1]. Replacing the membership functions of vertices and edges in fuzzy graphs by interval valued fuzzy sets such that they satisfy some particular conditions, interval valued fuzzy graphs (IVFGs) were defined. Thus IVFGs provide a better description of vagueness and uncertainty within the specific interval than the traditional fuzzy graph. The basic concepts of fuzzy sets, interval valued fuzzy sets and fuzzy graph theory can be found in [8], [19] and [18] to [23]. In 2009, Hongmei and Lianhua[6] introduced the definition of IVFG. Muhammad Akram and Wieslaw A. Dudek[1], defined the operations of cartesian product, composition, union and join on IVFGs and investigated some properties. They also introduced the notion of interval valued fuzzy complete graphs and obtained some properties of self complementary and self weak complementary interval valued fuzzy complete graphs. Muhammad Akram [2] also introduced interval valued fuzzy line graphs. A. A. Talebi and H. Rashmanlou [20] studied on isomorphism of IVFGs. H. Rashmanlou and Young Bae Jun[13] defined the three new operations, direct product, semi strong product and strong product of IVFGs and discussed their properties on complete IVFGs. Pradip Debnath[11] introduced domination in IVFGs. H. Rashmanlou and Madhumangal Pal defined irregular IVFG[9], balanced IVFG[14] and antipodal IVFG[15] and studied their properties. They also studied on the properties of highly irregular IVFG[17] and defined isometry on IVFG[16]. Muhammad Akram, Noura Omair Alshehri and W.A Dudek [3] introduced certain types of IVFG such as balanced IVFGs, neighbourly irregular IVFGs, neighbourly total irregular IVFGs, highly irregular IVFGs, highly total irregular IVFGs. Again Muhammad Akram, M. Murtaza Yousaf and W A Dudek [4] studied the properties of self centered IVFGs. Madhumangal Pal, Sovan Samanta and H. Rashmanlou [10] defined the degree and total degree of an edge in the cartesian product and composition of two IVFG and obtained some results. Basheer Ahammed Mohideen [5] studied on strong and regular IVFGs. S. Ravi Narayanan and N. R. Santhi Maheswari [18] introduced strongly edge irregular and strongly edge totally irregular IVFG and made a comparative study between the two. A. A. Talebi, H. Rashmanlou and Reza Ameri [21] discussed product IVFGs. K. Radha and M. Vijaya [12] discussed the totally regular property of the join of two fuzzy graphs. In this paper, we introduce and analyse the notion of TRIVFGs. Also we obtain some necessary and sufficient conditions for total regularity of IVFGs.