Dr.E.Chandrasekaran, N.Sathyaseelan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 3, May-Jun 2012, pp.733-738 733 | P a g e FUZZY NODE FUZZY GRAPH AND ITS CLUSTER ANALYSIS Dr.E.Chandrasekaran Associate Professor in Mathematics, Presidency College, Chennai. Tamilnadu, India . N.Sathyaseelan Assistant Professor in Mathematics, T.K Government Arts College, Vriddhachalam – 606 001. ABSRACT We could generally analyze inexact information efficiently and investigate the fuzzy relation by applying the fuzzy graph theory. We would extend the fuzzy graph theory and propose a fuzzy node fuzzy graph to the crisp node fuzzy graph by applying T-norms. In this paper, we would discuss about Fuzzy node fuzzy graph,fuzzy partial graph,Transformation from the fuzzy node fuzzy graph to the crisp node fuzzy graph, New T-norms, "sathyam product", Cluster structure analysis of fuzzy node fuzzy graph, And Global structure analysis of the optimal fuzzy graph G λ in the fuzzy sequence { G λ }. By using the fuzzy node fuzzy graph theory and this new T-norm, we would clarify the relational structure of fuzzy information and by using the decision of an optimal level on a partition tree. We could analyze the clustering relation among node. Keywords: crisp node fuzzy graph, fuzzy partial graph, fuzzy node fuzzy graph and quasi logical product. I. INTRODUCTION Fuzzy graph theory was introduced by Azriel Rosenfield in 1975. Though it is very young it has been growing fast and has numerous applications in various fields. In this simplest form a graph consists of set of elements (or nodes) and a set of ordered and unordered pairs of nodes (or edges). We would extend the fuzzy graph theory and propose a fuzzy node fuzzy graph. Since a fuzzy node fuzzy graph is complicated to analyse, we would transform it to a simplest fuzzy graph by using T-norms family. A graph is defined by a pair G = (V,F) where V={v 1 , v 2 ,….v n } is a finite set of vertices and F, a collection of edges that happen to connect these vertices. The edge set F graphically represented as V × V. The concept of fuzzy graph is the fuzzyfication of the crisp graphs using fuzzy sets. In this paper we introduce fuzzy node fuzzy graph, crisp node fuzzy graph, fuzzy partial graph and new T-norms (Triangular norms). Also we study some properties of fuzzy node fuzzy graphs and examine whether they hold for crisp node fuzzy graphs. II. PRELIMINARIES 2.1. FUZZY NODE FUZZY GRAPH Definition – 1: Crisp Node Fuzzy Graph The crisp node fuzzy graph G is defined by G = (V,F) : V = {v i }, F = {f ij }, 0  f ij  1. Where V is the set of nodes and F is an n × n matrix whose (i,j) component f ij is a fuzziness of the arc from the node v i to the node v j . Definition -2: Fuzzy Partial Graph If the fuzzy graph G = (V,F) and the fuzzy graph G' = (V,F') satisfies: F =(f ij ) , F' = ( ' ij f ), f ij  ' ij f . We call fuzzy graph G' as fuzzy partial graph of fuzzy graph g and denote G'πG. Definition -3: Fuzzy Node Fuzzy Graph A fuzzy node fuzzy graph G is defined by, G = (V, F) = {v i / u i }, Y = (y ij ), 0  u i  1, 0  y ij  1 Where V is the set of the nodes and the fuzziness u i is a fuzziness of the node v i . Y is an n  n matrix whose (i, j) component y ij is a fuzziness of the arc from the node v i to the node v j . A fuzzy node fuzzy graph is characterized by the fuzziness of the nodes and the fuzziness of the arcs. Therefore, the structure of a fuzzy node fuzzy graph is usually very complicated. Then, it should be interesting to transform a fuzzy node fuzzy graph to a crisp node fuzzy graph and we present a method to transform a fuzzy node fuzzy graph to a crisp node fuzzy graph.