IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-3008, p-ISSN:2319-7676. Volume 9, Issue 6 (Jan. 2014), PP 47-51 www.iosrjournals.org www.iosrjournals.org 47 | Page Irregular Intuitionistic fuzzy graph 1 A.Nagoor Gani, 2 R. Jahir Hussain, 3 S. Yahya Mohamed 1,2 P .G and Research Department of Mathematics, Jamal Mohamed College, Tiruchirappalli-620 020, India 3 Department of Mathematics , Govt. Arts College, Tiruchirappalli-620 022, India Abstract: In this paper, some types of Irregular intuitionistic fuzzy graphs and properties of neighbourly irregular ,highly irregular intuitionistic fuzzy graphs are studied. Some results on totally Irregular intuitionistic fuzzy graphs are established. Keywords: Intuitionistic fuzzy graph, degree, total degree, Intuitionistic fuzzy sub graph. 2010Mathematics Subject Classification: 03E72, 03F55 I. Introduction: Atanassov [1] introduced the concept of intuitionistic fuzzy (IF) relations and intuitionistic fuzzy graphs (IFGs). Research on the theory of intuitionistic fuzzy sets (IFSs) has been witnessing an exponential growth in Mathematics and its applications. This ranges from traditional Mathematics to Information Sciences. This leads to consider IFGs and their applications. R. Parvathy and M.G.Karunambigai’s paper [ 5] introduced the concept of IFG and analyzed its components. A. Nagoor Gani and S.R. Latha[3] introduced Irregular fuzzy graphs and discussed some of its properties. In this paper, some properties of Irregular Intuitionistic fuzzy graphs and neighbourly irregular intuitionistic fuzzy graphs are studied. Also Some results on totally Irregular intuitionistic fuzzy graphs are established. II. Preliminary Definition 2.1: A fuzzy graph G = (σ,µ) is a pair of functions σ : V → [ 0,1] and Definition 2.2: The fuzzy subgraph H = (τ, ρ) is called a fuzzy subgraph of G = (σ,µ) Definition 2.3: Let G = (σ,µ) be a fuzzy graph. The degree of a vertex v is d (v) = Σ u≠v μ (v,u) Definition 2.4: Let G = (σ,µ) be a fuzzy graph on G * : < V, E >. If d G (v) = k for all vV, i.e., if each vertex has the same degree k, then G is said to be a regular fuzzy graph of degree k or a k-regular fuzzy graph. Definition 2.5: Let G = (σ,µ) be a fuzzy graph on G*. The total degree of a vertex uV is defined by td G (u)= Σ u≠v μ(v,u) + σ(u) Definition 2.6: An Intuitionistic fuzzy graph is of the form G = < V, E > where (i) V={v1,v2,….,vn} such that μ 1 : V[0,1]and γ 1 : V [0,1] denote the degree of membership and nonmembership of the element vi V, respectively, and 0 ≤ μ 1 (vi) + γ 1 (vi) ≤ 1 for every vi V, (i = 1,2, ……. n), (ii) E  V x V where μ 2 : VxV[0,1] and γ 2 : VxV[0,1] are such that μ 2 (vi , vj) ≤ min [μ 1 (vi), μ 1 (vj)] and γ 2 (vi , vj) ≤ max [γ 1 (vi), γ 1 (vj) ] and 0 ≤ μ 2 (vi, vj) + γ 2 (vi,vj) ≤ 1 for every (vi ,vj) E, ( i, j = 1,2, ……. n) v1(0.2,0.7) (0.2,0.1) v2(0.3,0.5) (0.1,0.6) (0.3,0.5) (0.2,0.3) V4(0.5,0.2) (0.1,0.2) V3(0.4,0.3) Fig 1: Intuitionistic Fuzzy Graph