Table Of Content

Subset Vertex Graphs for Social Networks W. B. Vasantha Kandasamy Ilanthenral K Florentin Smarandache 2018 Copyright 2018 by EuropaNova ASBL and the Authors This book can be ordered from: EuropaNova ASBL Clos du Parnasse, 3E 1000, Bruxelles Belgium E-mail: [email protected] URL: http://www.europanova.be/ Peer reviewers: Prof. Dr. Tèmítópé Gbóláhàn Jaíyéolá, Department of Mathematics, Obafemi Awolowo University, Ile Ife 220005, Nigeria. Prof. Dr. Songtao Shao and Prof. Dr. Chunxin Bo, College of Information Engineering, Shanghai Maritime University, Shanghai 201306, China. Prof. Dr. Xiaohong Zhang, Department of Mathematics, School of Arts and Sciences, Shaanxi University of Science & Technology, Xi’an 710021, China. Many books can be downloaded from the following Digital Library of Science: http://fs.gallup.unm.edu/ScienceLibrary.htm ISBN-13: 978-1-59973-538-2 EAN: 9781599735382 Printed in the United States of America 2 CONTENTS Preface 4 Chapter One INTRODUCTION 7 Chapter Two SUBSET VERTEX GRAPHS OF TYPE I 21 Chapter Three SUBSET VERTEX GRAPHS OF TYPE II AND THEIR APPLICATIONS 183 FURTHER READING 279 3 INDEX 286 ABOUT THE AUTHORS 288 4 PREFACE In this book authors for the first time introduce the notion of subset vertex graph using the vertex set as the subset of the power set P(S), S is assumed in this book to be finite; however it can be finite or infinite. We have defined two types of subset vertex graphs, one is directed and the other one is not directed. The most important fact which must be kept in record is that for a given set of vertices there exists one and only one subset vertex graph be it of type I or type II. Several important and innovative features of these graphs are defined, described and developed in this book. The set S considered can be real or complex or indeterminate or dual numbers depending on the problem in hand. However by the definitions of the subset vertex graphs of type I and type II the edges or relation between two vertices are always real. 5 Now based on the property that in case of subset vertex graphs for a given set of vertices the graph is unique, these graphs are well suited for social networking. This is one of the special features enjoyed by these subset vertex graphs. The applications of these new types of subset vertex graphs are dealt in chapter three of this book. Authors suggest several open problems for any researcher. Finally these new types of subset vertex graphs will be very useful in general to researchers in computer networks and in particular to researchers in social information networks. We wish to acknowledge Dr. K Kandasamy for his sustained support and encouragement in the writing of this book. W.B.VASANTHA KANDASAMY ILANTHENRAL K FLORENTIN SMARANDACHE 6 Chapter One INTRODUCTION Social information networks concept was introduced or perceived by researchers Emile Durkheim and Ferdinand Tonnies as social groups as early as 1890’s [1, 17]. However Tonnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and beliefs or impersonal, formal and instrumental social links but Durkheim gave a non individualistic explanation of social facts arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel analyzed the network size on interaction and examined and likelihood of interaction in loosely knit networks rather than groups [13]. Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology and mathematics working independently [47]. Jacob L. Moreno [9-11] has studied by systematic recording and analysis of social interaction in small groups, like class rooms and work groups. 8 Subset Vertex Graphs for Social Networks He was the one who defined and developed sociometry a quantitative method for measuring social relationship. He studies the relationship between social structures and psychological wellbeing. “Sociometric explorations reveal the hidden structures that give a group its form: the alliances, the subgroups, the hidden beliefs, the forbidden agendas, the ideological agreements the ‘starts of the show’ [47]. One of Moreno’s innovations in sociometry was the development of the sociogram, a systematic method for graphically representing individuals as points/nodes and the relationships between them as lines/arcs. Moreno who wrote extensively of his thinking, applications and findings also founded a journal sociometry. For more about the development please refer [9-11]. Finally just for the better understanding of the reader, Facebook is a social network service and website which is largely based on the sociometry of its users. In this chapter some basic properties of graphs are recalled so as to make this book a self-contained one. It is well known when we have G = (V, E) the graph, then V is the set of vertices v , v , …, v and E = {e / 1  i, j  n} are edges of the 1 2 n ij graph G. If e in G exists we call them as loops or a graph with self ii connection of vertices. We will just provide some examples. Example 1.1: Let G = (V, E) be the graph given by the following figure. Introduction 9 v v 1 2 v 5 v 3 v 4 v 6 Figure 1.1 V = {v , v , v , v , v , v } is the vertex set of the graph G and 1 2 3 4 5 6 {e , e , e , e , e , e , e , e } = E is the edge set. As the 12 25 56 46 34 13 45 16 graph is not directed we have every e  E; e = e . 1  i, j  8. ij ij ji Study in this direction can be had from [3]. Further if there is no direction given in edges we call it as a graph and is not a directed graph. Further this graph G has no loops. We give the matrix associated with G which is as follows. v v v v v v 1 2 3 4 5 6 v 0 1 1 0 0 1 1 v 1 0 0 1 1 0 2 M = v 1 0 0 1 0 0 . 3 v 0 1 1 0 1 0 4 v 0 1 0 1 0 1 5 v 1 0 0 0 1 0 6 We see the matrix is a square matrix with main diagonal entries to be zero. Clearly M is symmetric about the diagonal.