### On Edge Regular Fuzzy Soft Graphs

B. Akilandeswari
2019 International Journal for Research in Applied Science and Engineering Technology
In this paper, degree of an egde and total degree of an edge in a fuzzy soft graph is introduced, edge regular fuzzy soft graphs, and totally edge regular fuzzy soft graphs are also introduced . Theorems for edge regular fuzzy soft graphs and totally edge regular fuzzy soft graphs are introduced. A necessary condition under which they are equivalent is provided. Some properties of edge regular fuzzy soft graphs and totally edge regular fuzzy soft graphs are studied. Keywords: Degree of an edge
more » ... Degree of an edge in a fuzzy soft graph, total degree of an edge in a fuzzy soft graph, edge regular fuzzy soft graph , totally edge regular fuzzy soft graph. I. INTRODUCTION In 1736, Euler first introduced the concept of graph theory. In the history of mathematics, the solution given by Euler of the wellknown Konigsberg bridge problem is considered to be the first theorem of graph theory. The graph theory is a very useful tool for solving combinatorial problems in different areas such as operations research, optimization, topology, geometry, number theory, algebra and computer science. Fuzzy set theory, introduced by Zadeh in 1965 is a mathematical tool for handling uncertainties like vagueness, ambiguity and imprecision in linguistic variables  . Research on theory of fuzzy sets has been witnessing an exponential growth; both within mathematics and in its application. Fuzzy set theory has emerged as a potential area of interdisciplinary research and fuzzy graph theory is of recent interest. The first definition of fuzzy graph was introduced by Haufmann in 1973, based on Zadeh's fuzzy relations in 1971. In 1975, Rosenfeld introduced the concept of fuzzy graphs  . Nagoor Gani and Latha  introduced irregular fuzzy graphs. The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs using fuzzy relations, obtaining analogs of several graph theoretical concepts. During the same time Yeh and Bang have also introduced various connectedness concepts in fuzzy graph  . Now, fuzzy graphs have been witnessing a tremendous growth and finds application in many branches of engineering and technology. A. NagoorGani and K. Radha introduced the concept of regular fuzzy graphs in 2008  . In 1999, D.Molodtsov introduced the notion of soft set theory to solve imprecise problems in economics, engineering and environment. He has shown several applications of this theory in solving many practical problems. There are many theories like theory of probability, theory of fuzzy sets, theory of intuitionistic fuzzy sets, theory of rough sets, etc. which can be considered as mathematical tools to deal with uncertainties. But all these theories have their own inherent difficulties. The theory of probabilities can deal only with possibilities. The most appropriate theory to deal with uncertainties is the theory of fuzzy sets, developed by Zadeh in 1965. But it has an inherent difficulty to set the membership function in each particular r cases. Also the theory of intuitionistic fuzzy set is more generalized concept than the theory of fuzzy set, but also there have same difficulties. The soft set theory is free from above difficulties. In 2001, P.K.Maji, A.R.Roy,R.Biswas [20, 21] initiated the concept of fuzzy soft sets which is a combination of fuzzy set and soft set. In fact, the notion of fuzzy soft set is more generalized than that of fuzzy set and soft set. Muhammad Akram and Saira Nawaz  introduced more concepts on fuzzy soft graphs. K. Radha and N. Kumaravel  introduced new concepts based on edge regular fuzzy soft graph.