Vulnerability parameters in fuzzy graphs

Abstract

Connectivity plays a vital role in graph theory. Removal of some vertex subset, makes a graph disconnected into some connected components. It is necessary to have a parameter to relate that vertex subset and connected components. Integrity, a vulnerability parameter deals with a vertex subset and maximum component after removing that vertex subset. There is no such parameter defined in fuzzy graph so far. Chapter 1 presents basic definitions of graph theory and fuzzy graph theory with examples. It also consolidates the literature survey of fuzzy graph theory. In chapter 2, integrity (vertex integrity) of fuzzy graphs is defined. It deals with all the standard fuzzy graphs such as path, cycle, complete graph and bipartite graph. Integrity value of fuzzy graph is discussed with certain conditions, like vertex membership values are constant, edge values are constant, having particular node strength sequence. This chapter concludes with integrity of Cartesian product, join and union operator of fuzzy graphs. In chapter 3, edge integrity of fuzzy graph is defined and discussed with detailed examples. Generally, vertex integrity deals only with vertex membership values, which include both edge and vertex membership values. Basic results for underlying crisp graph path, cycle, star, complete graph, bipartite graph is found. Edge integrity of fuzzy graphs with certain condition are also found and the comparison of results between vertex integrity and edge integrity is presented To define the connectedness, a path is enough in a graph. Using this concept, connectedness of a fuzzy graph is defined in chapter 4. A new vulnerability parameter, span integrity, is defined with basic results. A brain network has been modelled as fuzzy graph. EEG data are converted into time series data and then this data is converted into edge membership value of fuzzy graph. Span integrity of these fuzzy graphs are found to analyze the effect of meditation in brain regions. Domination theory is applied in many real time problems. Dominating sets are considered as integrity sets. With this vertex subset, in chapter 5, domination integrity of fuzzy graphs is defined and explained with examples and basic results. A new class of fuzzy graph, efficient fuzzy graph, is introduced. A real time problem model is taken and analysed for this parameter. Power domination is used in PMU placement in electrical network. Power dominating sets are not unique. Power domination integrity of crisp graph is not yet found in literature. In chapter 6, a new algorithm is described and is used to find the number of PMUs to observe the entire network. This algorithm is tested in IEEE 14, IEEE 30, IEEE 57 standard bus systems. As a result of this test, more than one power dominating sets are found. Domination integrity parameter is defined and applied in the above standard buses to choose the optimum. newline