Pseudo Lattice Graphs and their Applications to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Florentin Smarandache Ilanthenral K 2014 This book can be ordered from: EuropaNova ASBL Clos du Parnasse, 3E 1000, Bruxelles Belgium E-mail: [email protected] URL: http://www.europanova.be/ Copyright 2014 by EuropaNova ASBL and the Authors Peer reviewers: Dr. Stefan Vladutescu, University of Craiova, Romania. Dr. Octavian Cira, Aurel Vlaicu University of Arad, Romania. Mumtaz Ali, Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan Said Broumi, University of Hassan II Mohammedia, Hay El Baraka Ben M'sik, Casablanca B. P. 7951. Morocco. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/eBooks-otherformats.htm ISBN-13: 978-1-59973-296-1 EAN: 9781599732961 Printed in Europe and United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two PSEUDO LATTICE GRAPHS OF TYPE I 9 Chapter Three PSEUDO LATTICE GRAPHS OF TYPE II 99 Chapter Four PSEUDO NEUTROSOPHIC LATTICE GRAPHS OF TYPE I AND TYPE II 179 3 Chapter Five SUGGESTED PROBLEMS 223 FURTHER READING 261 INDEX 274 ABOUT THE AUTHORS 276 4 PREFACE In this book for the first time authors introduce the concept of merged lattice, which gives a lattice or a graph. The resultant lattice or graph is defined as the pseudo lattice graph of type I. Here we also merge a graph with a lattice or two or more graphs which call as the pseudo lattice graph of type II. We merge either edges or vertices or both of a lattice and a graph or a lattice and a lattice or graph with itself. Such study is innovative and these mergings are adopted on all fuzzy and neutrosophic models which work on graphs. The fuzzy models which work on graphs are FCMs, NCMs, FRMs, NRMs, NREs and FREs. This technique of merging FCMs or other fuzzy models is very advantageous for they save time and economy. Moreover each and every expert who works on the problems is given equal importance. 5 We called these newly built models as merged FCMs, merged NCMs, merged FRMs, merged NRMs, merged FREs and merged NREs. We wish to acknowledge Dr. K Kandasamy for his sustained support and encouragement in the writing of this book. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE ILANTHENRAL K 6 Chapter One INTRODUCTON In this chapter we just give some of the properties enjoyed by graphs and lattices. For in this book we obtain new classes of lattice-graphs by merging two lattices or by merging a lattice and a graph or a graph and a graph. We use the term merging as follows. We may merge a vertex of a lattice L with another vertex or L or a edge and two 1 2 vertices of a lattice L with an edge and two vertices of a lattice 1 L or merge many vertices and many edges of a lattice L with 2 1 that of a lattice L . 2 Such study is new and innovative. It goes without saying that every lattice is a connected graph but a graph in general is not a lattice, for v v  2 1 v 3 v5   v 4 8 Pseudo Lattice Graphs and their Applications to Fuzzy… is a graph and not a lattice. Further v 1 v8   v 2 v  9 v7 v5  v3 v  10 v4  v 6 is a graph and not a lattice. Now when in a lattice L merged by a vertex or edge or 1 both with another lattice L we get the resultant graph which is 2 termed as a pseudo lattice graph of type I it may be a lattice or a graph. Similarly using a lattice and a graph or a graph and a graph we get a graph termed as the pseudo lattice graph of type II. This notion finds its applications in fuzzy and neutrosophic models which work on direct graphs like FCMs, NCMs, FRMs, NRMs, FREs and NREs [79, 90]. This book also studies about merging of neutrosophic lattices [87]. Thus these new type of merging may also find more applications in due course of time. Several open problems are suggested. 8 Chapter Two PSEUDO LATTICE GRAPHS OF TYPE I In this chapter we introduce a new mode of construction of graphs using lattices. We take two lattices merge one vertex or two vertices or three vertices or so on or merge one edge and two vertices or more edges and more vertices and arrive at a diagram. The resultant can be a graph or a lattice. We will first illustrate all these situations by some examples. Example 2.1: Let L be the chain lattice C 1 7 1   a 1 a 2 a3 a 4  a 5  0