Types of uncertain nodes in a fuzzy graph.

The graph theory has numerous applications in the problems of operations research, economics, systems analysis, and transportation systems. However, real applications of a graph theory are full of linguistic vagueness, i.e., uncertainty. For example, the vehicle travel time or number of vehicles on a road network may not be known precisely. The fuzzy graph model can be used to model the complex, not clearly explained uncertain real life applications, in which conventional graph may fail to model properly. In a fuzzy graph, it is very important to identify the nature (strength) of nodes and no such analysis on nodes is available in the literature. In this paper, we introduce a method to find out the strength of the node in a fuzzy graph. The degree of the node and maximum membership value of the adjacent edges of that node are used to compute the strength of the node. The strength of a fuzzy node itself is a fuzzy set. Depending upon the strength of the nodes, we classify the nodes of a fuzzy graph into six types namely α strong fuzzy node, β strong fuzzy node, regular fuzzy node, α weak fuzzy node, β weak fuzzy node and balance fuzzy node.