Vertex connectivity parameters paths and cycles in fuzzy graphs

Abstract

One of the major developments in the 20th century is the evolution of very large interconnection networks. Even the life of a common man is controlled by several such networks. Among different models representing a network, a graph structure is the most feasible one. In such a model, vertices represent objects and edges represent links between them. Designing of a network that is ideal from all perspectives is almost impossible. Depending on the requirement, a nearly suitable one can be designed. When we have a large network, its dynamics can be explained only by its local behavior. A comparison of the performances between different regions can be done only by introducing a fuzzy graph. When there is an uncertainty regarding the capacities, then also a fuzzy graph model is relevant. As far as fuzzy graphs are concerned, the concept of connectivity is very crucial, as the real world networks are ideally related to them. The term connectivity can be translated into different terms like maximum bandwidth, maximum width, maximum deliverable speed, bottleneck capacity, etc. based on the type of network we discuss. There are several connectivity parameters, using which one can evaluate the performance of a network. For example, a higher value for the average bandwidth in internet network is necessary, for its better performance and stability. The main objectives of this thesis is to study some of the important connectivity parameters related to a network, represented as a fuzzy graph and to characterize fuzzy graph theoretical structures like fuzzy trees, fuzzy cycles and complete fuzzy graphs using them. The motivation for this study comes especially from applications related to human trafficking and illegal immigration. Vertex connectivity is the major theme of this work. Average vertex connectivity, t-connected and uniformly connected fuzzy graphs are also studied in detail.