Abstract
Graph models are fundamental in network theory. But normalization of weights are necessary to deal with large size networks like internet. Most of the research works available in the literature have been restricted to an algorithmic perspective alone. Not much have been studied theoretically on connectivity of normalized networks. Fuzzy graph theory answers to most of the problems in this area. Although the concept of connectivity in fuzzy graphs has been widely studied, one cannot find proper generalizations of connectivity parameters of unweighted graphs. Generalizations for some of the existing vertex and edge connectivity parameters in graphs are attempted in this article. New parameters are compared with the old ones and generalized values are calculated for some of the major classes like cycles and trees in fuzzy graphs. The existence of super fuzzy graphs with higher connectivity values are established for both old and new parameters. The new edge connectivity values for some wider classes of fuzzy graphs are also obtained. The generalizations bring substantial improvements in fuzzy graph clustering techniques and allow a smooth theoretical alignment. Apart from these, a new class of fuzzy graphs called generalized t-connected fuzzy graphs are studied. An algorithm for clustering the vertices of a fuzzy graph and an application related to human trafficking are also proposed.
Highlights
The 20th century witnessed several major revolutions in mathematics
Most of the concepts in the recently introduced fuzzy graph theory can be directly applied to problems which were previously developed algorithmically
This article is an attempt to redefine some of the existing connectivity parameters
Summary
The 20th century witnessed several major revolutions in mathematics. The invention of fuzzy logic by Zadeh [1] in 1965 is a remarkable one with several implications and amazing consequences. We rectify this problem by providing new definitions for vertex connectivity and edge connectivity in fuzzy graphs. The first paper in fuzzy graph theory by Rosenfeld [2] addressed the problem of clustering using the concept of vertex connectivity. Yeh and Bang in Reference [4] simultaneously They studied vertex connectivity and edge connectivity of fuzzy graphs and used them in fuzzy graph clustering. In 2009, Mathew and Sunitha [14] identified different types of edges in fuzzy graphs and provided an algorithm for the same They characterized many fuzzy graph structures like fuzzy trees, blocks and complete fuzzy graphs in an effective way using this identification. Fuzzy vertex connectivity and fuzzy edge connectivity were introduced in 2010, by the authors of Reference [3] They were generalizations of Yeh and Bang’s connectivity parameters.
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