W. B. Vasantha Kandasamy Ilanthenral K Florentin Smarandache. MOD Graphs

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2 W. B. Vasantha Kandasamy lanthenral K Florentin Smarandache MOD Graphs

3 Many books can be downloaded from the following Digital Library of Science: SBN-: EAN: Printed in the United States of America

4 MOD Graphs W. B. Vasantha Kandasamy lanthenral K Florentin Smarandache 0

5 This book can be ordered from: EuropaNova ASBL Clos du Parnasse, E 000, Bruxelles Belgium info@europanova.be URL: Copyright 0 by EuropaNova ASBL and the Authors

6 CONTENTS Preface 7 HSTORY OF NEUTROSOPHC THEORY AND TS APPLCATONS ABOUT THE BOOK Chapter One BASC CONCEPTS 5 Chapter Two MOD GRAPHS 7 Chapter Three MOD NATURAL NEUTROSOPHC GRAPHS AND THER PROPERTES 9 Chapter Four MOD BPARTTE GRAPHS 7 FURTHER READNG NDEX 5 ABOUT THE AUTHORS 8 5

7 Peer reviewers: Professor Paul P. Wang, Ph D, Department of Electrical & Computer Engineering, Pratt School of Engineering, Duke University, Durham, NC 7708, USA Dr.S.Osman, Menofia University, Shebin Elkom, Egypt. Said Broumi, University of Hassan Mohammedia, Hay El Baraka Ben M'sik, Casablanca B. P. 795, Morocco Florentin Popescu, Facultatea de Mecanica, University of Craiova, Romania.

8 PREFACE History of Neutrosophic Theory and its Applications Zadeh introduced the degree of membership/truth (t) in 95 and defined the fuzzy set. Atanassov introduced the degree of nonmembership/ falsehood (f) in 98 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/ neutrality (i) as independent component in 995 (published in 998) and defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood): Etymology. The words neutrosophy and neutrosophic were coined/ invented by F. Smarandache in his 998 book. Neutrosophy: A branch of philosophy, introduced by F. Smarandache in 980, which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Neutrosophy considers a proposition, theory, event, concept, or entity, "A" in relation to its opposite, "Anti-A" and that which is not A, "Non-A", and that which is neither "A" nor "Anti-A", denoted by "Neut-A". Neutrosophy is the basis of neutrosophic logic, neutrosophic probability, neutrosophic set, and neutrosophic statistics. {From: The Free Online Dictionary of Computing, edited by Denis Howe from England. Neutrosophy is an extension of the Dialectics.} Neutrosophic Logic is a general framework for unification of many existing logics, such as fuzzy logic (especially intuitionistic fuzzy logic), paraconsistent logic, intuitionistic logic, etc. 7

9 The main idea of NL is to characterize each logical statement in a D-Neutrosophic Space, where each dimension of the space represents respectively the truth (T), the falsehood (F), and the indeterminacy () of the statement under consideration, where T,, F are standard or non-standard real subsets of ] - 0, + [ with not necessarily any connection between them. For software engineering proposals the classical unit interval [0, ] may be used. T,, F are independent components, leaving room for incomplete information (when their superior sum < ), paraconsistent and contradictory information (when the superior sum > ), or complete information (sum of components = ). For software engineering proposals the classical unit interval [0, ] is used. For single valued neutrosophic logic, the sum of the components is: 0 t+i+f when all three components are independent; 0 t+i+f when two components are dependent, while the third one is independent from them; 0 t+i+f when all three components are dependent. When three or two of the components T,, F are independent, one leaves room for incomplete information (sum < ), paraconsistent and contradictory information (sum > ), or complete information (sum = ). f all three components T,, F are dependent, then similarly one leaves room for incomplete information (sum < ), or complete information (sum = ). n general, the sum of two components x and y that vary in the unitary interval [0, ] is: 0 x + y - d (x, y), where d (x, y) is the degree of dependence between x and y, while d (x, y) is the degree of independence between x and y. n 0 Smarandache refined the neutrosophic set to n components: (T, T,...;,,...; F, F,...); see PiP.pdf. 8

10 The Most mportant Books and Papers in the Development of Neutrosophics introduction of neutrosophic set/logic/probability/statistics; generalization of dialectics to neutrosophy; (last edition) 00 introduction of neutrosophic numbers (a+b, where = indeterminacy) 00 introduction of -neutrosophic algebraic structures 00 introduction to neutrosophic cognitive maps introduction of interval neutrosophic set/logic 00 introduction of degree of dependence and degree of independence between the neutrosophic components T,, F (p. 9) ndence.pdf 007 The Neutrosophic Set was extended [Smarandache, 007] to Neutrosophic Overset (when some neutrosophic component is > ), since he observed that, for example, an employee working overtime deserves a degree of membership >, with respect to an employee that only works regular fulltime and whose degree of membership = ; and to Neutrosophic Underset (when some neutrosophic component is < 0), since, for example, an employee making more damage than benefit to his company deserves a degree of membership < 0, with respect to an employee that produces benefit to the company and has the degree of membership > 0; 9

11 and to and to Neutrosophic Offset (when some neutrosophic components are off the interval [0, ], i.e. some neutrosophic component > and some neutrosophic component < 0). Then, similarly, the Neutrosophic Logic/Measure/Probability/Statistics etc. were extended to respectively Neutrosophic Over-/Under-/Off- Logic, Measure, Probability, Statistics etc. Offset.pdf Smarandache introduced the Neutrosophic Tripolar Set and Neutrosophic Multipolar Set and consequently the Neutrosophic Tripolar Graph and Neutrosophic Multipolar Graph (p. 9) introduction of N-norm and N-conorm development of neutrosophic probability (chance that an event occurs, indeterminate chance of occurrence, chance that the event does not occur) ty.pdf 0 - refinement of components (T, T,...;,,...; F, F,...) 0

12 0 introduction of the law of included multiple middle (<A>; <neuta>, <neuta>, ; <antia>) development of neutrosophic statistics (indeterminacy is introduced into classical statistics with respect to the sample/population, or with respect to the individuals that only partially belong to a sample/population) introduction of neutrosophic precalculus and neutrosophic calculus 05 refined neutrosophic numbers (a+ b + b + + b n n ), where,,, n are subindeterminacies of indeterminacy ; 05 (t,i,f)-neutrosophic graphs; 05 - Thesis-Antithesis-Neutrothesis, and Neutrosynthesis, Neutrosophic Axiomatic System, neutrosophic dynamic systems, symbolic neutrosophic logic, (t, i, f)-neutrosophic Structures, - Neutrosophic Structures, Refined Literal ndeterminacy, Multiplication Law of Subindeterminacies: 05 ntroduction of the subindeterminacies of the form n k 0 0, for k {0,,,, n-}, into the ring of modulo integers Z n - called natural neutrosophic indeterminacies [Vasantha-Smarandache]

13 05 ntroduction of neutrosophic triplet structures and m- valued refined neutrosophic triplet structures [Smarandache - Ali] Submit papers on neutrosophic set/logic/probability/statistics to the international journal Neutrosophic Sets and Systems, to the editor-in-chief: smarand@unm.edu ( see )

14 ABOUT THE BOOK n this book authors for the first time introduce study and develop the notion of MOD graphs, MOD directed graphs, MOD finite complex number graphs, MOD neutrosophic graphs, MOD dual number graphs and so on using edge weights from Z n, C(Z n ) Z n, Z n g and so on. Likewise MOD directed natural neutrosophic graphs are defined. Further type, type and type. MOD directed graphs and MOD natural neutrosophic graphs are defined and developed. This book has over 85 examples and over 50 figures. The notion of MOD bipartite graphs and MOD natural neutrosophic bipartite graphs using Z, C (Z n ) Z n and so on are described. This book gives the probable applications of these concepts to MOD mathematical models like MOD Cognitive Maps model and MOD Relational Maps model which have been introduced by the authors. There are open conjectures which can help the researchers in graph theory. Several innovative results are obtained. We wish to acknowledge Dr. K Kandasamy for his sustained support and encouragement in the writing of this book. n W.B.VASANTHA KANDASAMY LANTHENRAL K FLORENTN SMARANDACHE

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16 Chapter One BASC CONCEPTS n this book for the first time authors venture to study MOD graphs using Z n, Z n, C(Z n ), Z n, Z n g, C (Z n ), Z n h and so on. MOD graphs take vertex set and (or) edge sets from any of the sets Z n, C(Z n ), Z n, Z n g, Z n h and Z n k. These MOD graphs are special for these lead to MOD Cognitive Maps model [8]. So an exhaustive study of MOD graphs is carried out in this book. Next we study MOD natural neutrosophic graphs and directed graphs with edge weights / vertex sets from Z or Z n h or Z n or C (Z n ) or Z n g or Z n k. Such study is thoroughly carried out in this book. These graphs find applications in the study of MOD natural neutrosophic Cognitive Maps model [8]. So a systematic study is made in this book. For the first time we visualize edges and vertex sets to be natural neutrosophic, natural neutrosophic dual numbers, natural neutrosophic-neutrosophic edges / vertices and so on. Such n 5

17 study is only new and innovative but can find application in MOD Cognitive Maps models. [7-5, 8]. Next we proceed onto introduce and newly describe the new notion of MOD bipartite graphs and MOD n-partite graph with edge / vertex set from any one of the sets Z n, Z n g, C(Z n ), Z n h, Z n,z n k. These structures find applications in MOD Relational Maps model with edge weights from Z n or Z n g or Z n or C(Z n ) or Z n h or Z n k [9]. These models will be new for edge weights / vertex sets can be complex or dual numbers or neutrosophic or special dual like numbers or special quasi dual numbers. So such study is not only new and innovative but is very useful. Next we study of MOD n-partite graphs with vertex sets / edge sets from Z n or C(Z n ) or Z n or Z n h or Z n g or Z n k. We now proceed onto describe MOD natural neutrosophic bipartite graph with edge weights / vertex sets from Z n or Z or C (Z n ) or Z n h or Z n g or Z n k. n These MOD natural neutrosophic bipartite graphs can find applications in MOD natural neutrosophic Relational Maps model [9]. The edge weights can be from Z n h or Z n k or Z n g. Zn or C (Z n ) or Z n or Such study is new and innovative for we can have the nodes to be natural neutrosophic or complex or dual number or special quasi dual number or special dual like numbers. For more about these concepts refer [7-50, 55].

18 Chapter Two MOD GRAPHS n this chapter we for the first time introduce the notion of MOD graphs and MOD directed graphs. A MOD graph is a graph where the vertex sets is either a subset of Z n or whole of Z n ( n < ). MOD directed graphs are of three types. n type MOD directed graphs the vertex set can be any thing but the edge weights are from Z n ; n <. n type MOD directed graphs both vertices as well as edge weights are from Z n. n type MOD directed graphs vertices are from Z n but edge weights from the set {0, }. The general MOD graphs are graphs whose vertex sets are from Z n or subsets of Z n may not be directed. We will first provide some examples of each of the situations and also suggest problems. 7

19 Example.: Let {G} be the MOD graphs with vertex set from Z = {0, }. {} {} { } Figure. are the only four MOD graphs using the vertex set Z. Example.: Let {G} be the MOD graphs with vertex set for Z = {0,, }. {} {} {} { } { } 0 0 { } { } { } { } { } o Figure. 8

20 There are 7 MOD graphs using the vertex set Z = {0,, }. Example.: Let {G i } be the collection of all MOD graphs using vertex set from Z = {0,,, }. {0}, {}, {}, {}, {, } {0, }, {0, }, {0, }, {, }, {, }, {0,, }, {0,, } {0,, }, {,, }, {0,,, },

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22 and so on. Figure.

23 So even in case of Z we see getting all the MOD graphs happens to be a challenging problem. We take {0,,, } = Z as the vertices {v 0, v, v, } for one can easily work with vertices for it is non abstract. We see when we work with MOD integers Z n they can be applied to semi automaton, automaton or in networking. We leave open the following conjecture. Conjecture.: Let {G i } be the collection of all MOD graphs with vertex set from Z n ; n <. Find the number of such MOD graphs which take its vertices from subsets of Z n or Z n ; n <. We have provided examples of them. This is introduced mainly for appropriate applications. n case of MOD graphs the elements of Z n can be given face value ordering or the vertices can be given face values which would be useful in case of networking or semi automaton or automation. For the associated face values for their vertices can predict the importance or otherwise of these vertices from Z n ; n <. Next we proceed onto describe type MOD directed graphs by examples. Example.: Let G be the MOD directed graph with edge weights from Z 5 and v, v,, v 7 are the vertices of G.

24 v v G = v v 7 Figure. The MOD type matrix M associated with G is as follows: M = v v v v v v v 5 7 v v v v v v v Example.5: Let V be a MOD directed graph with edge weights from the subset of Z. v,v,, are the vertices associated with V.

25 v 0 v V = 5 5 Figure.5 The MOD type matrix S associated with V is as follows: v v v v v 5 v v S =. v v v The edge weights of graph V are only from a subset of Z. These types of MOD directed graphs have been already used in MOD Cognitive Maps model [8]. We have some advantages of using these type MOD graphs. For we see we can find S, S and so on.

26 v v v v v 5 v Now S v = v v v v v v v v 5 v v v v v = v v v v v 5 v v v v v is associated with a MOD directed graph with edge weights from Z but has loops. S The graph associated with S is as follows. v v Figure. 5

27 Now we find S = v v v v v 5 v v v v v v v v v v 5 v v v v v = S S = v v v v v 5 v v v v v The type MOD directed graph associated with S follows: is as v v 9 Figure.7 The type MOD directed graph associated with S has no loops.

28 Consider S = S S = v v v v v 5 v v v v v v v v v v 5 v v v v v = v v v v v 5 v v v v v The graph related to S conjecture only the following: has two loops. Thus we can Conjecture.: Let G be type MOD directed graph with related adjacency matrix M. Edge weights of G are from Z n. Characterize those type MOD graph G so that. i) The type MOD directed graph H related with M have always loops (specify under what conditions it will have no loops). ii) Can type MOD directed graph P related with M be free of loops? 7

29 iii) Characterize those MOD directed graph related to say odd powers of matrix M say M t+ will have no loops and that of even powers M t will have loops. Next we proceed onto describe one more type MOD directed graph G and the related MOD adjacency matrix by an example. Example.: Let G be the type MOD directed graph with edge weights from Z 7 given by the following figure with vertex set v, v,,, and v with no loops. G = v v 5 v Figure.8 The related type MOD adjacency matrix M related with G is as follows: 8

30 M = v v v v v v 5 v v v v v v v v v v v v 5 v v M = v v v v The type MOD graph related with M has two loops. Now we find M in the following M = v v v v v v 5 v v v v v v This the type MOD directed graph associated with the adjacency matrix M has three loops. 9

31 We see the type MOD directed graph matrix behaves in such a way so that the following conjecture is made. Conjecture.: Characterize those type MOD directed graphs G so that their squares, cubes, etc. represented by their MOD type matrices i) has no loops, ii) always has loops. t is pertinent to keep on record that these type MOD directed graphs with edge weight from Z n, have already been applied to MOD Cognitive Maps model [8]. Thus they will find applications in mathematical modelling. Next we proceed onto describe type MOD directed graphs. These type MOD directed graphs take both edge values as vertex sets from subsets of Z n. We will describe this situation by some examples. Example.7: Let G be a MOD directed graph with vertices v, v,, v 7 from Z 0 and edge weights from the set Z 0 given by the following figure G = 8 Figure.9 This G is a type MOD directed graph. The type MOD adjacency matrix M associated with G is as follows: 0

32 M = The following rules is to be compulsorily followed to avoid confusion. We know there is a face value ordering in Z 0 also 0 is the least and 9 is a greatest so the vertex with v =, v =, =, = 5, =, v = 7 and v 7 = 8. Thus we have the vertices arrange according to the face value ordering in Z n. We will give one more example of type MOD directed graph in the following. Example.8: Let H be the type MOD directed graph with vertices and edge weights from Z 5 given by the following figure. 5 8 H = Figure.0

33 The type MOD adjacency matrix N of the graph H is as follows: N = Now we can adopt this MOD directed graph of type for automaton, semi automaton and networking apart from mathematical modeling. We proceed onto enumerate the properties enjoyed by the MOD type matrices and their related graphs. Example.9: Let G be the type MOD directed graph given by the following figure with edge weights and vertex set from Z 8. G = v 0 v 5 Figure.

34 (v =, v = 7, = 0, = 5 and = ). G. Let M be the type MOD adjacency matrix associated with M = v v v v v 5 v v v v v We find M M = M = v v v v v 5 v v v v v n the type MOD directed graph associated with MOD matrix M we see the MOD graph G has two loops and has more edges connected; G is as follows: G = v 5 0 v Figure. 0

35 Next we find the value of M and the corresponding type MOD directed graph G. M = v v v v v 5 v v 0 v v 0 0 v Clearly the type MOD directed graph G is as follows: G = v v 0 0 Figure. The number of loops have increased. The number of edges has increased. Some of the weights of the directed edges has also increased. Next we find M and the related type MOD directed graph G.

36 v v v v v 5 v M v = v 0 0. v 0 v 5 The type MOD directed graph has loops and 5 directed weighted edges. The MOD directed type graph is as follows. G = v v Figure. Consider M 9 = v v v v v 5 v v 0 v 0 0. v 0 0 v 5 There are only three loops. 5

37 Let G 9 be the type MOD directed graph given by the following figure. G 9 = v v v Figure.5 Thus we cannot say anything about this type MOD directed graph. We see as we product it if we choose to call so then it is clearly seen there is increase in directed edges. We conclude this notion with one more example by taking a small value of n for Z n. Example.0: Let G be the type MOD directed graph which is as follows with edge weights from Z. v v v = 0, =, =, = 5, v =

38 M = Figure. v v v v v 5 v v v v v The type MOD adjacency matrix. We find M, M, M, M and their related type MOD graphs. M = v v v v v 5 v v v v v The type MOD directed graph associated with M be G which is as follows. v v v Figure.7 7

39 This MOD directed graph of type has seven edges of which two are just loops. We now find M in the following. M = v v v v v 5 v v v v 0 0 v The type MOD directed graph G related with M is as follows : v G = v Figure.8 Clearly G has no loops only six edges. The edges has reduced for seven to six. Now we find M ; 8

40 v v v v v 5 v M v 0 0 = v v v Let G be the type MOD directed graph given by M. This has two loops and 8 weighted edges given by the following figure. G v v Figure.9 We know find M 5, M 5 = v v v v v 5 v v v 0 0 v v Let G 5 be the type MOD directed graph of M 5 given by the following figure. 9

41 G 5 = v v Figure.0 Let M be the MOD matrix of type. M = v v v v v 5 v v v v v The type MOD directed graph G associated with M is as follows: v G = v Figure. The graph has only edges and four loops. 0

42 Finally before we comment on this graph G find M 7. M 7 = v v v v v 5 v v v v v Clearly the type MOD directed graph G 7 associated with M 7 has 9 edges and no loops is given below. G 7 = v v Figure. M 8 = v v v v v 5 v v v v v 5 0 0

43 This type MOD directed graph G 8 has two loops and six edges given by the following figure. G 8 = v v Figure. Now we find M 9, M 9 = v v v v v 5 v v v v 0 0 v Thus the type MOD directed graph associated with M 9 has only edges and no loops. The graph G 9 is as follows. G 9 = v Figure. v

44 From this graph it is easily seen all graphs G n+ have no loops whereas all G n has loops. Study in this direction is left as an exercise to the reader. Next we proceed onto describe type MOD directed graphs by examples. Example.: Let G be the type MOD directed graph given by the following figure. The vertices take their values from Z 7. G = v v where v = 0, v =, =, =, =, v = 5 and v 7 =. Figure.5 Let M be the type MOD adjacency matrix related to G v 7 v M = v v v v v v v 5 7 v v v v v v v

45 We find M M = v v v v v v v 5 7 v v v v v v v v v v v v v v 5 7 v v v v v v v The type MOD directed graph associated with M be G which is as follows: G = v v v Figure. v 7

46 This is the way special product operation is performed. n the usual product operation M is kept as it is v v v v v v v 5 7 v v M v =. v v v v Here there are two methods apart from the special one which thresholds all values greater than one to one. Other one keeps the value as it is as long as the values are in mod 7 as vertex set is from Z 7. So if 8 occurs in M t then it will be and so on (t ). Yet another type of operation is the expert wishes to take weight from any Z n ; n < and the product is performed. We will describe each by an example. Example.: Let G be the type MOD directed graph with edge weights from {0, } and vertex set from Z given by the following figure: 5

47 v v G = v = 0, v =, =, = and =. Figure.7 The type MOD matrix B of G is as follows: B = v v v v v 5 v v v v v Now we find B B = v v v v v 5 v v v v v The MOD type directed graph B is as follows:

48 v B = v Figure.8 We find B B = v v v v v 5 v v v v v The MOD type directed graph B is as follows. B = v v Figure.9 7

49 v v v v v 5 v Let B v = v v v The type MOD directed graph B represented by M is as follows. B = v v Figure.0 This has only one edge and four loops. Consider B 5 v v v v v 5 v B 5 = v v v v Let B 5 be the MOD directed type graph given by the following figure: 8

50 v v Figure. Clearly the type MOD directed graph has no loops. We find B B = v v v v v 5 v v v v v Clearly the type MOD directed graph has no loops. v v Figure. 9

51 This we see after a finite number of iterations say some k iterations we will get B k = B. This type MOD directed graph behaves in a very different way. Next we proceed onto describe MOD graphs with vertex sets from subsets of Z n h or C(Z n ) or Z n g or Z n h or Z n k. This study is not only new but also relevant for at times the vertex set can be imaginary or indeterminate or a dual number or a special dual like number or a special quasi dual number. So to cater to these needs these new types of MOD graphs are most important. We call MOD graph to be a MOD neutrosophic graph if the vertex sets are subsets of Z n = {a + b / a, b Z n, = }. We will provide some examples of such graphs. Example.: Let G be the MOD neutrosophic graph with vertex set from Z 0 = {a + b / a, b Z 0, = } given by the following figure: 50

52 G = v 7 v 7 v v v Figure. Now there are situations in machines as well as in networking where the nodes can be indeterminate at one stage (repair or over used or heated or low power) in case of machines and (in mathematical modeling where nodes can be indeterminate) respectively. Example.: Let G be the MOD neutrosophic graph with vertex weights from the set Z given by the following figure : 5

53 0 v v + v + v 7 + v 9 + G v 8 Figure. Next we proceed onto describe MOD finite complex number graphs by some examples. We call a MOD graph which takes the vertex set values from C(Z n ) = {a + bi F / a, b Z n, i F = (n )} are defined as MOD complex graphs or MOD finite complex number graphs. We will illustrate this situation by some examples. Example.5: Let G be the MOD complex graph with vertex set from C(Z ) given by the following figure: 5

54 +i F 0 v i F v G = +i F i F+ v 9 v Figure.5 Example.: Let H be the MOD finite number complex graph with edge weights from C(Z ) which is given by the following figure: H = Figure. 5

55 These MOD graphs will find its applications in mathematical modeling when the nodes are imaginary or mixed imaginary or real. Next we proceed onto describe MOD dual number graphs. f a MOD graphs takes its vertex set values from the set Z n g = {a + bg / a, b Z n, g = 0} then we define the MOD graph as MOD dual number graph. We will describe this situation by some examples. Example.7: Let H be the MOD dual number graph given by the following figure with vertex set from Z 9 g. g 0 g +5g 7 8 +g H = 5g Figure.7 5

56 Example.8: Let V be the MOD dual number graph with edge weights from Z g given by the following figure: V = g+ g g+ 0 g g+ Figure.8 These newly constructed MOD graphs can find lots of applications in various fields. All the more MOD dual number graphs can be very helpful when the nodes are mixed dual numbers or dual numbers or real values. Next we describe MOD special dual like number graphs. Let G be a MOD graphs if the vertex set is from Z n h = {a + bh / a, b Z n, h = h} then we define G to be a MOD special dual like number graph. We will describe this by some examples. Example.9: Let B be the MOD special dual like number graph with vertex elements from Z 7 h given by the following figure: 55

57 +h h B = 5h +h h +h Figure.9 Example.0: Let V be the MOD special dual like number graph with vertex set from Z h given by the following figure: 5h 0 +7h V = h+ 9 0h h +h +h Figure.0 5

58 Example.: Let G be a MOD graph with vertex set from Z 5 k = {a + bk / a, b Z 5 and k = k} given by the following figure : v k k+ v G = v 9 v 8 8k 9k+ 7 v k v Figure. G will be known as the MOD special quasi dual number graph. Thus if G is a MOD graph which takes vertex sets from; Z n k = {a + bk / a, b Z n, k = (n ) k} then we define G to be a MOD special quasi dual number graph. We will give one more example of this situation. 57

59 Example.: Let H be the MOD special quasi dual number graph with vertex elements from Z k given by the following figure: 0k 0 k +5k 7k k H = k k Figure. These MOD graphs will also find appropriate applications in mathematical modeling and so on. Next we proceed onto describe type MOD neutrosophic graphs, type MOD dual number graphs, type MOD complex number graphs and so on only by examples. 58

60 A MOD directed graphs which has any vertex set but whose edge weights are from Z n are defined as type MOD neutrosophic directed graph. Example.: Let V be the type MOD neutrosophic directed graph with edge weights from Z given by the following figure: V = v v v + Figure. The adjacency matrix M associated with V is as follows: v v v v v v 5 v v M = v v v v We find 59

61 M = v v v v v v 5 v v v v v v The type MOD neutrosophic directed graph associated with M be V which is as follows: v v V = + v Figure. Now we find M = v v v v v v 5 v v v v v v

62 The type MOD directed neutrosophic graph is given by v v V = + v Figure.5 This has no edge and no loops. Example.: Let S be the type MOD neutrosophic directed graph with edge weights from Z 5 given by the following figure: S= + v v v Figure. The type MOD neutrosophic matrix P associated with S is as follows:

63 v v v v v v 5 v v P = v v v v We now find the square of P in the following: P = v v v v v v 5 v v v v v v The type MOD neutrosophic directed graph S associated with P is as follows: S = v + v Figure.7 v

64 Next we find P. v v v v v5 v v v P = v v v v The type MOD neutrosophic directed graph S is as follows: S v v v Figure.8 S has three loops all of them are pure neutrosophic. Edge weights of S are also pure neutrosophic. Next we find P in the following:

65 v v v v v v 5 v v P = v v v v The type MOD directed neutrosophic graph S related with P is as follows. S = v v Figure.9 Now we find P 5 = v v v v v v 5 v v v v v v Let S 5 be the type MOD directed neutrosophic graph associated with P 5 which is as follows:

66 v S 5 = v v Figure.50 Thus we can find any number such MOD directed neutrosophic graphs of type for a given MOD directed graph. This will have certainly some implications in mathematical modeling as well as it would also can suggest a model given by one expert say M is related to another experts model on the same problem as M = M t, (t > 0). Such study can also relate the experts opinion in a distinct and innovative way. Another problem in this direction is can we say if S is the MOD type neutrosophic matrix related with the MOD type directed graph, then M n = (0) for some n or M n = M? Study in this direction is new and left as an exercise to the reader. Next we proceed onto define and describe MOD directed finite complex number graphs of type. 5

67 Let G be a type MOD directed graph if the edge weights are from C(Z n ) then we define G to be a type MOD finite complex number directed graph. We will illustrate this situation by some examples. Example.5: Let G be the type MOD directed finite complex number graph with edge weights from C(Z ) given by the following figure. G = v +i F v i F i F Figure.5 Let M be the type MOD finite complex matrix associated with G. M = v v v v v 5 v v 0 0 if 0 0 v if 0. v 0 if v We find M

68 v v v v v 5 v 0 0 if if 0 M v if 5 0 = v v 0 0 if 0 0 v if 0 0. Let G be the type directed MOD finite complex number graph associated with M given by the following figure: v v G = +i F i F +5 i F i F + 5+i F Figure.5 Now we find M M = v v v v v 5 v if 5 0 v 0 if v 0 0 if 0 0. v if 0 v i F 5iF 0 7

69 The MOD directed type graph G associated with M is as follows. v v +i F +i F 5+i F +i F G = 5i F + +i F Figure.5 This type MOD finite complex directed graph has three loops and edges. We now find M, M = v v v v v 5 v 0 if v 0 0 if 0 0 v if 0. v 0 if v 5 0 if 0 if 0 We give the type MOD finite complex number directed graph G associated with M in the following: 8

70 G = +i F v v i v F +i F +i F +i F i F + Figure.5 We now find M 5 M 5 = v v v v v 5 v 0 0 if 0 0 v v 0 if v v if 0 0 The type MOD directed graph of finite complex numbers G 5 associated with M 5 is as follows: v v G 5 = i F +i F i F Figure.55 9

71 Thus we can find any M n and its associated G n type MOD finite number directed graph. Next we proceed onto describe type MOD directed dual number graphs. Let G be a type MOD directed graph withi edge weights from Z n g = {a + bg / a, b Z n, g = 0}, we call G to be the type MOD directed dual number graph. We will illustrate this situation by some examples. Example.: Let G be the type MOD directed dual number graph with edge weights from Z 8 g given by the following figure. v v g g+5 +g G = v 5g v 7 g Figure.5 Let M be the type MOD dual number matrix associated with G; 70

72 v v v v v v v 5 7 v v g v g M =. v v g v 0 g v g 0 We find M M = v v v v v v v 5 7 v g v g 0 0 v g 0 0. v g v v g v 7 0 g The type MOD directed dual number graph G associated with M is as follows. v v G = g +g +g 7g v g g Figure.57 v 7 7

73 Next we find M in the following M = v v v v v v v 5 7 v g 0 0 v g v g v v v g 0 0 v The type MOD directed dual number graph G is as follows. G = v +g g v Figure.58 g v v 7 g This type MOD dual number directed graph G has only four edges no loops. M = v v v v v v v 5 7 v g v v v v v v

74 The type MOD dual number directed graph G is as follows. v v G = g v 7 v g Figure.59 So at one stage we will have M n = (0) for some finite n, (n > 0). nterested reader can work more such type MOD dual number directed graphs. Now we proceed onto define type MOD special dual like number directed graphs. A type MOD directed graph if it takes its edge weights from Z n h = {a + bh / a, b Z n, h = h} is defined as the type MOD directed special dual like number graph. We will illustrate this situation by some examples. Example.7: Let G be the MOD type special dual like number directed graph with edge weights from Z 0 h. The following figure for G is given below: 7

75 G = 5 v v h+ v v 8 8h v 7 +9h Figure.0 The associated type MOD matrix of G is as follows: N = v v v v v v v v v v h v v v h v v v h 0 Now we proceed onto find N. Let the corresponding type MOD graph associated with N be G. 7

76 v v v v v v v v v v h v N = v h. v h 0 v v v h 0 0 The type MOD special dual like number graph associated with N is as follows. v v +8h G = 8h v v 7 h v h Figure. Now we find N in the following: 75

77 v v v v v v v v v v h v N = v h 0. v h 0 0 v h v v8 0 0 h 8h The type MOD special dual like number directed graph G associated with N is as follows: G = v v +h +8h h v h 8h +8h v 8 h v 7 Figure. There is no loops only weighted edges. Likewise we can find the type MOD directed graph with edge weights from Z 0 h. We now give one example of the type MOD directed graph G with edge weights from 7

78 Z n k = {a + bk / a, b Z n, k = (n )k}, the graph G will also be known as the type MOD special quasi dual number graph. Example.8: Let G be the type MOD directed special quasi dual number graph with edge weights from Z 9 k = {a + bk / a, b Z 9, k = 8k}. The figure of G is as follows: 8k v v +k 5+k v v 7 +k Figure. The type MOD matrix of G is as follows: M = v v v v v v v 5 7 v 0 8k v v v 0 0 k v k v k v

79 We give M in the following: M = v v v v v v v 5 7 v k 0 v k v 0 5k v k v k v k 0 0 v k The type MOD directed graph G is as follows. G = v 5k +k v v 5+5k 5k 5+k +k v 7 8+k Figure. Now we find M in the following: 78

80 v v v v v v v 5 7 v k v k 0 0 M v k 0 =. v v 5 k v k v k The MOD type directed graph G is as follows: G = +k v v 8+k k 5+5k k v +5k v 7 +5k Figure.5 Now we find M = v v v v v v v 5 7 v k 0 0 v k v k. v v v 0 0 k v 7 k

81 The type MOD special quasi dual number directed graph G associated with M is as follows. G = v +k +k v k +k v 7k v 7 Figure. This is the way the product operation is performed using type MOD special quasi dual number directed graphs with edge weights from Z n k = {a + bk / k = (n )k; a, b Z n }. We now leave it for the reader to develop the properties of MOD directed graphs built using various sets like C(Z n ) or Z n or Z n g or Z n h or Z n k and analyse the special feature associated with them. We suggest the following problems for the interested reader. Problems. Let G be the MOD graph with entries from Z 7. i) How many such MOD graphs can be got using Z 7? ii) Find the number of MOD graphs using Z n, ( n < ). iii) What are the special features enjoyed by these MOD graphs? 80

82 . Let G be the MOD directed graph given by the following figure with vertex the following figure with vertex set from Z 7. v v G = v v 7 Figure.7 i) Find all MOD graphs isomorphic with G. ii) All MOD graphs with seven vertices not isomorphic with G. iii) Find all MOD graphs (distinct) with seven vertices. iv) How many MOD graphs with six vertices from Z 7 can be constructed? v) Study question (iv) for 5 and vertices. vi) Find the number of MOD graphs with three vertices from Z 7.. Study any other distinct feature associated with MOD graphs.. Let G be the MOD neutrosophic graph with edge set from Z n. i) Show all MOD graphs are included in the MOD neutrosophic graphs. ii) Find the number of distinct MOD neutrosophic graphs with Z n number of vertices. 8

83 iii) Find the number of MOD neutrosophic graphs with 5 vertices. iv) Enumerate all special features associated with MOD neutrosophic graphs. 5. What are the special and distinct features enjoyed by MOD finite complex number graphs with vertex set from C(Z n )?. Show certainly these can find many application as we tred over finite number of vertices. 7. Study MOD dual number graphs with vertex set from Z n g. Show this will have lot of application when one works with dual number as vertices. 8. Let {G} be the collection of all MOD special dual like number graph with vertex set from Z h or subsets of Z h. i) How many graphs exist in {G}? ii) Does these graphs enjoy any special property? iii) How many of these MOD special dual like number graphs with vertex set from Z h are complete graphs? 9. Let B = {collection of all MOD special quasi dual number vertex set graphs with vertex sety from Z 0 k or subset of {Z 0 k}. i) Find o(b). ii) How many are complete MOD graphs? iii) Compare the collection when Z 0 k is replaced by a) Z 0, b) Z 0 g, 8

84 c) Z 0 and d) C(Z 0 ). 0. What are the special features associated with type MOD directed graph?. Let G = {collection of all type MOD directed graphs with edge weights from subsets of Z 9 or Z n }. i) Find o(g). ii) Hence find o(g) if Z 9 is replaced by Z n, n <. iii) f G is given by the following figure: v G = 7 8 v 5 v Figure.8 a) Find the type MOD connection matrix M associated with G. t b) Find M, M, M,., M ; t < and the corresponding type MOD graphs. c) Can we say M n = (0) or M n = M t after a finite number of products t = or or?. Distinguish between type MOD directed graphs and MOD graphs.. Let {G} be the collection of all type MOD directed neutrosophic graphs with edge weights from Z 8. 8

85 a) Find o({g}). b) Let H {G} given by the following figure: v v + + Figure.9 v 7+ f M is the type MOD matrix find M, M and so on and the related H, H and so on. c) What type of M t ; t < have loops? d) When will M s = (0); s <? e) Find G {G} which has 0 vertices taking edge weights from Z 8.. Let {G} be the collection of all type MOD finite complex number directed graph with edge weights from C(Z ). a) Find o({g}). b) f H {G} is given by the following figure: G = +i F v v i F 8+5i F 7i F Figure.70 v 5+8i F 8

86 f M is the MOD associated matrix find the type MOD graphs associated with M, M and so on. c) Can we say M t = M or M s = (0)? 5. Let {H} be the collection of all type MOD finite dual number directed graph with edge weights from Z g. i) Find o({h}). ii) f H be a MOD finite dual number directed graph of type by the following figure: +g v v H = g 5g +5g v g 7+g v 7 Figure.7 f M is the MOD matrix dual numbers associated with H find M, M and so on and obtain the corresponding MOD type graphs. iii) Which of the MOD type graphs are free from loop? iv) Enumerate all type MOD directed graphs which has loops.. Let {P} be the collection of all MOD type directed special dual like number graph with edge weights from Z h = {a + bh / a, b Z, h = h}. a) Find o({p}). b) f G be a graph in {P} with 7 vertices how many type MOD directed special dual like number graphs can be obtained. 85

87 c) How many G i s are there? d) Let H = +h 7 v 8+h v +h h v 8+0h v 7 v 8 5 v 9 Figure.7 be the type MOD directed special dual like number graph. M the associated type MOD matrix. Find M, M, the corresponding MOD directed special dual like number graphs, which of them have loop? 7. Let {G} be the collection of type MOD special quasi dual number directed graphs with edge weights from Z k = {a + bk / k = k, a, b Z }. i) Find o({g}). ii) Let V be the type special quasi dual number MOD directed graph given by the following figure: 8

88 V = v 5k +k v k k v 0k v 7 Figure.7 Find the type MOD directed special quasi dual number MOD matrix M of V. iii) Find M, M and M 7 and their respective MOD type directed special quasi dual number graphs V, V and V 7. iv) Which of these graphs have loops? v) Describe any other special feature associated with these type MOD directed special quasi dual number graphs. vi) Compare this with type MOD directed graphs, type MOD directed dual number graph and type finite complex number graph. 8. Describe and develop type MOD directed graphs. 9. Let {G} be the collection of all MOD dual number directed graphs of type with edge weights from Z 8 g. i) Find o({g}). ii) How many of the type MOD directed dual number graphs will have loops? iii) Enumerate all special features enjoyed by type MOD directed dual number graphs. iv) Compare type MOD directed dual number graph with type MOD directed dual number graphs. 87

89 0. Study questions (i) to (iv) of problem 9 in case of type MOD directed neutrosophic graphs with edge weights from Z 9.. Study questions (i) to (iv) of problem 9 in case of type MOD directed finite complex number graphs with edge weights from C(Z ).. Describe all special features associated with type MOD directed graphs using Z n or Z n or Z n g or Z n h or Z n k or C(Z n ).. Distinguish MOD type directed graphs from MOD type directed graphs and MOD type graphs.. Let G be the type MOD directed graph G with vertex set from Z and edge sets from {0, } given by the following figure. G = v v v Figure.7 i) Find the type MOD matrix M associated with G. ii) Find M and the related graph G. Does G have loops? iii) Can we say there exist a n such that M n = M? iv) s it possible M t = (0) for t <? v) Which is true in this case (iii) or (iv)? 5. Let G be the type MOD dual number directed graph given by the following figure with edge weights from Z g. 88

90 v v g+8 5g+ 8 5g v 8 v 5g g v 7 +g v 8 Figure.75 i) Find M related with G. ii) Find M, M, M 7 and M 9 and the related graphs G, G, G 7 and G 9 respectively. iii) Can M t ( t < ) have loops? iv) What is the smallest t so that M t has loops? v) Can M t = (0) for some t, t <? vi) Can M t = M for some t, t <? vii) Enumerate any other special and interesting feature enjoyed by this type MOD directed dual number graph G.. Let G be the type MOD directed finite complex number graph given by the following figure with edge weights from C(Z ). i F v v +i F 5+i F v +i F Figure.7 v 7 v 8 i F 89

91 i) Find the MOD type finite complex number matrix M associated with G. ii) Find M, M, M 9 and M and the related MOD type directed finite complex number graphs. iii) Which M t has loops? iv) Find the smallest t so that G t has loops. v) Will odd order M n+ or even order M n contribute to MOD type graphs with loops? vi) Will M t = (0) or M t = M? 7. Let V be the type MOD directed graph with edge weights from Z given by the following figure: v 0 v 9 v v 7 v 8 Figure.77 i) Find M the type MOD matrix associated with V. ii) Find M, M, M 8, M and M and the corresponding V, V, V 8, V and V respectively. iii) For what power of M the relation type MOD directed graph has loops? iv) Can M t = (0)? v) Can M t = M or M s = (0), ( t, s < )? 90

92 Chapter Three MOD NATURAL NEUTROSOPHC GRAPHS AND THER PROPERTES n this chapter for the first time we introduce the notion of MOD natural neutrosophic graphs in a systematic way. However in [8] we have used this concept in the MOD natural neutrosophic Cognitive Maps model. Further in this book we use zero dominant MOD natural t neutrosophic product that is 0. m = 0; t = n or g or h or c or or k, m Z n is a zero divisor or nilpotent or an idempotent. We will proceed onto describe this notion first by examples. Example.: Let G be the MOD natural neutrosophic graph with vertex set from subsets of Z or whole of Z by the following figures. 9

93 Figure. There are several MOD natural neutrosophic graphs using Z. nfact finding the number of MOD natural neutrosophic graphs with vertex set Z happens to be challenging problem. Conjecture.: Let be the MOD natural neutrosophic set. Finding the total number of MOD natural neutrosophic graphs happens to be a challenging one. 9

94 Example.: Let Z be the MOD natural neutrosophic set. + 0 G = Figure. The MOD natural neutrosophic graph with vertex set from Z is given in Figure. G = 0 Figure. is a MOD natural neutrosophic graph with two vertices. 0 + G = + 0 Figure. 9

95 0 G = 0 Figure.5 G, G, G and G are the MOD natural neutrosophic graphs with vertex set from Z. Example.: Let G be the MOD natural neutrosophic graph with entries from Z given by the following figures: = H Figure. 0 0 H = Figure.7 9

96 Next we can construct MOD natural neutrosophic finite complex number graph with vertex set from C (Z n ). This will be described by the following examples. Example.: Let G be the MOD natural neutrosophic finite complex numbers with vertex set from C (Z ). + 0 i F i F 8i F i F = K Figure i F K = Figure.9 K and K are MOD natural neutrosophic finite complex number graphs with vertex set from C (Z n ). 95

97 Example.5: Let G be the MOD natural neutrosophic finite complex number graph with vertex set from C (Z 5 ) i F i F Figure.0 Next we just give an example or two of MOD natural neutrosophic dual number graphs in the following. A MOD graph which takes its vertex set from the MOD natural neutrosophic dual number set Z n g = {a + bg / a, b Z n, g = 0} will be known as the MOD natural neutrosophic dual number graph. Example.: Let G be the MOD natural neutrosophic dual number graph with vertex set from set Z 9 g. G is given by the following figure: 8 g g g 8g 5g+ g 5g g g 0 Figure. 9

98 We next give one more example of MOD neutrosophic dual number graphs. natural Example.7: Let G be the MOD natural neutrosophic dual number graph with vertex set from Z g given by the following figure: g g 0 + g g g g g g 0 g g 0 g Figure. When we need labeling differently these MOD graphs will play a vital role. For the labeled graphs can get the labeling from C (Z n ) or Z n g or Z n or Z k or Z h. Z n or We can also obtain the adjacency matrix of a labeled graph. Thus both MOD graphs and MOD natural neutrosophic graphs can take the vertex values or distinctly labeled as per need. 97