VOL. 10, NO. 12, JULY 2015 ISSN 1819-6608 ARPN Journal of Engineering and Applied Sciences © 2006-2015 Asian Research Publishing Network (ARPN). All rights reserved. www.arpnjournals.com 5413 INTUITIONISTIC DOUBLE LAYERED FUZZY GRAPH J. Jesintha Rosline and T. Pathinathan Department of Mathematics, Loyola College, Chennai, India E-Mail: jesi.simple@gmail.com ABSTRACT In fuzzy graph theory, double layered fuzzy graph and intuitionistic fuzzy graph have been defined already by different authors. In this paper Intuitionistic double layered fuzzy graph is defined with examples. Some of its theoretical concepts were studied using different concepts in IFG. Keywords: double layered fuzzy graph, order of IFG, size of IFG, vertex degree of IFG. 1. INTRODUCTION In 1965, Zadeh published his seminal paper on “Fuzzy Sets” which described fuzzy set theory and consequently fuzzy logic [12]. The purpose of Zadeh’s paper was to develop a theory which could deal with ambiguity and imprecision of certain classes of sets in human thinking, particularly in the domains of pattern recognition, communication, information and abstraction. Azriel Rosenfeld in 1975 introduced the notion of fuzzy graph and several fuzzy analogs of graph theoretic concepts such as path, cycles, connectedness etc. [5]. Zadeh in 1987 introduced the concept of fuzzy relations. Mordeson in 1993 introduced the concepts of fuzzy line graphs and developed its basic properties. Sunitha and Vijayakumar discussed about the operations of union, join Cartesian product and composition on two fuzzy graphs [4]. The degree of a vertex in some fuzzy graphs was discussed by Nagoorgani and Radha [6]. Nagoorgani and Malarvizhi have defined different types of fuzzy graphs and discussed its relationships with isomerism in fuzzy graphs [3]. The degree of a vertex in some fuzzy graphs was introduced by A. Nagooor Gani and K. Radha [2]. The first definition of intuitionistic fuzzy relations and intuitionistic fuzzy graphs were introduced by Atanassov in 1986[16] and further studied in 2009[13]. The operations on IFG was introduces by R. Parvathi, M. G. Karunambigai and K. Atanassov [13]. Degree, Order and Size in IFG was introduced by A. NaggorGani and S. ShajithaBegum[14]. The double layered fuzzy graph was introduced by T. Pathinathan and J. Jesintharosline, they have examined some of the properties of DLFG [1]. The vertex degree of cartesian product of intuitionistic fuzzy graph is given by T. Pathinathan and J. Jesintharosline [14]. In this paper Intuitionistic double layered fuzzy graph is defined and illustrated with some examples. Some of its properties is also analysed using intuitionistic fuzzy graph concepts and operations. First we go through some of the basic definitions. 2. PRELIMINARIES a) Fuzzy graph [4] A fuzzy graph G is a pair of functions G: (σ,µ) where σ is a fuzzy subset of a non empty set S and µ is a symmetric fuzzy relation on σ . The underlying crisp graph of G: (σ,µ) is denoted by * * * :( , ) G   . b) Intuitionistic fuzzy graph (IFG) [12] An IFG is of the form G: (V, E) where (i) V = {v1, v2, v3, … ,vn} such that 1 μ :V [0,1] ® and 1 γ :V [0,1] ® denote the degree of membership and non - membership of the element i v V Î respectively, and 1 i 1 i 0 μ (v ) γ (v ) 1 £ + £ for every i v V,(i 1,2,...n) Î = (1) (ii) E V V Í ´ where 2 μ :E [0,1] ® and 2 γ :E [0,1] ® are such that 2 i j 1 i 1 j μ (v , v ) μ (v ) μ (v ) £ Ù (2) 2 i j 1 i 1 j γ (v ,v ) γ (v ) γ (v ) £ Ú (3) And 2 i j 2 i j 0 μ (v v ) γ (v v ) 1 £ + £ (4) for every i j (v ,v ) E,(i, j = 1,2,...,n). Î Notations The triplets i 1i 1i (v , μ , γ ) denotes the degree of membership and non – membership of the vertex vi. The triple ij 2ij 2ij (e , μ , γ ) denotes the degree of membership and non – membership of the edge relation ij i j e = (v , v ) on V. c) Vertex degree of IFG [13] Let G = (V,E) be an IFG. Then the degree of a vertex v is defined by μ γ d(v) (d (v), d (v)) = where μ 2 u v d (v) = μ (u,v) ¹ å and γ 2 u v d (v) = γ (u,v) ¹ å .