Study on parameters related to connectivity and cycle connectivity of fuzzy graphs

Abstract

Graph models are fundamental in network theory. But normalization of weights are newlinenecessary to deal with large size networks like internet. Most of the works available newlinein the literature have been restricted to algorithmic perspective alone. In this area not newlinemuch have been studied theoretically on connectivity of normalized networks. Fuzzy newlinegraph theory answers to most of the problems. Although the concept of connectivity newlinein fuzzy graphs has been widely studied, one cannot find proper generalizations of newlinegraph connectivity parameters to fuzzy graphs. newlineThe main objective of this thesis is to study different connectivity parameters newlinerelated to such fuzzy graphs. We characterize fuzzy graph theoretical structures newlinelike fuzzy trees, fuzzy cycles, and complete fuzzy graphs. We also propose a newlineproper generalization for the existing connectivity parameters. New parameters are newlinecompared with the old ones and generalized values are obtained for some of the newlinemajor classes of fuzzy graphs. This generalization brings substantial improvements newlinein fuzzy graph clustering techniques and allow a smooth theoretical alignment. Apart newlinefrom these, a new class called generalized t-connected fuzzy graphs are also studied. newlineCyclic reachability is a novel concept related to the dynamics of a network. newlineCyclic connectivity determines cyclic reachability, in terms of strong cycles available newlinein the network. Different aspects of cyclic connectivity are discussed in this thesis. newlineSome new results related with cycle connectivity have been found in different nonisomorphic newlinefuzzy graph structures. Also, a study on cyclic boost vertices and cyclic newlineboost edges have been made and are characterized in a sequence of results. As newlinecycle connectivity is defined between a pair of vertices in a fuzzy graph, we reconceptualized newlinecycle strengths associated with a vertex and termed it as cycle cogency newlineof a vertex. Related algorithms are also obtained.Towards the end, we attempt to study a new connectivity parameter in fuzzy newlinegraphs, termed as connectivity remainder.