A study on irregular intuitionistic fuzzy graphs of second type

: In this paper, we deﬁne the Irregular and Complement of intuitionistic fuzzy graphs of second type. Also we establish some of their properties.


Introduction
Fuzzy sets were introduced by Lotfi A. Zadeh [10] in 1965 as a generalisation of classical (crisp) sets. Further the fuzzy sets are generalised by Krassimir T. Atanassov [1] in which he has taken non-membership values also into consideration and introduced Intuitionistic Fuzzy sets (IFS) and their extensions like Intuitionistic Fuzzy sets of second type (IFSST), Intuitionistic L-Fuzzy sets (ILFS) and Temporal Intuitionistic Fuzzy sets (TIFS). R. Parvathi and N. Palaniappan [5] introduced some Operations on IFSST and A. Shannon and K. T. Atanassov [6] discussed the theory of Intuitionistic Fuzzy Graphs. R. Parvathi and M. G. Karunambigai [4] introduced Intuitionistic Fuzzy Graphs (IFG) and elaborately and analyzed various components. Further A. Nagoor Gani and S. Shajitha Begum [2] introduced the concepts of degree, regular and irregular IFG.
In this paper we further study the intuitionistic fuzzy graphs of second type and some of their properties. In Section 2, we give some basic definitions and in Section 3, we define the irregular and complement of intuitionistic fuzzy graphs of second type. Also establish some of their properties. The paper is concluded in Section 4.

Preliminaries
In this section, we give some basic definitions. 1] denote the degree of membership and non-membership of the element v i ∈ V , respectively, and Definition 2.12. [8] Let G = [V, E] is an IFGST then the minimum degree of G is denoted by δ(G) and defined as, is an IFGST then the maximum degree of G is denoted by ∆(G) and defined as, Definition 2.14. [9] An IFGST G = [V, E] is said to be regular, if every vertex adjacent to vertices with same degree.

Irregular and complement of intuitionistic fuzzy graphs of second type
In this section, we define the irregular, complement of intuitionistic fuzzy graphs of second type and establish some of their properties.     Therefore a highly irregular IFGST need not be a neighbourly irregular IFGST. Proof. Let G = [V, E] be an IFGST with v 1 , v 2 , . . . , v n ∈ V . Assume that G is highly irregular IFGST and neighbourly irregular IFGST.
Therefore the degrees of all vertices of G are distinct. Conversely, assume that the degrees of all vertices of G are distinct. Which implies that, every two adjacent vertices have distinct degrees and to every vertex, the adjacent vertices have distinct degrees. That is, G is neighbourly irregular IFGST and highly irregular IFGST. This completes the proof of the theorem.      Therefore, the complement of a neighbourly irregular IFGST need not be neighbourly irregular.

Conclusion
In this paper, we have defined the irregular, complement of intuitionistic fuzzy graphs of second type. Also established some of their properties. In future, we will study some more properties and applications of IFGST.