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...This book has four chapters. Chapter one is introductory in nature. The reader is expected to have a good background of algebra and graph theory in order to derive maximum understanding of this research. The second chapter represents groups as graphs. The main feature of this chapter is that it contains 93 examples with diagrams and 18 theorems. In chapte...

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...Preface 5 Chapter One INTRODUCTION TO SOME BASIC CONCEPTS 7 1.1 Properties of Rooted Trees 7 1.2 Basic Concepts 9 Chapter Two GROUPS AS GRAPHS 17 Chapter Three IDENTITY GRAPHS OF SOME ALGEBRAIC STRUCTURES 89 3.1 Identity Graphs of Semigroups 89 3.2 Special Identity Graphs of Loops 129 3.3 The Identity...

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...th, for instance. At the vegetative level, we are affected through plants and trees; at the animate degree, through animals and through our own bodi... ... elements of the ego and the phases to correct each of them in detail. Using graphs, charts, and calculations, Kabbalah leads the individual throug... ...mals behave in this manner, and we shall briefly review the leading ones. The theory of “Group Selection” asserts that altruism serves the good of t... ...ch an animal be- longs, hence the specific animal is rewarded by it, too. The theory of “Kin Selection” explains that if altruism is turned toward t... ... it indirectly contrib- utes to the survival of its own genes. The “Symbiosis” theory argues that altruistic behavior is based upon the particular a...

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...d now I had a chance to treat my life as a living laboratory and put this theory to the test. I resolved to be easy on myself and to approach the fu... ...and designs, and in that session we came up with a final decision for the graphics. One of our goals was to have the graph­ ics reflect the friendlin... ...incorporating the phrase "Baked With Aloha" and adding the image of palm trees, which Christine designed. Time was so short that we didn't have the...

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...introduction of neutrosophic theory has put forth a significant concept by giving representation to indeterminates. Uncertainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is modeled we use f...

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... 1.2 Neutrosophic Vector spaces 8 Chapter Two SOME BASIC RESULTS ON GRAPH THEORY AND THEIR APPLICATION TO FUZZY MODELS 2.1 Some basics... ...eutrosophic Vector spaces 8 Chapter Two SOME BASIC RESULTS ON GRAPH THEORY AND THEIR APPLICATION TO FUZZY MODELS 2.1 Some basics on Gr... ...GRAPH THEORY AND THEIR APPLICATION TO FUZZY MODELS 2.1 Some basics on Graphs 18 2.2. More Properties on Graphs 26 2.3 Directed Graphs 32 2... ...Graphs 18 2.2. More Properties on Graphs 26 2.3 Directed Graphs 32 2.4 Trees 37 2.5 Independent – Sets and matchings 40 2.6 Eulerian and Ham... ... 37 2.5 Independent – Sets and matchings 40 2.6 Eulerian and Hamiltonian Graphs 44 2.7 Graph Colorings 52 2.8 Application of Graphs to Fuzzy M... ...2 3.3. Neutrosophic regular graphs and its properties 76 3.4. Neutrosophic Trees and Neutrosophic Eulerian graphs 84 3.5. Neutrosophic Graph Col... ...ophic algebraic structures is very recent. The introduction of neutrosophic theory has put forth a significant concept by giving representation to i... ...n almost all real-world problems. When uncertainty is modeled we use fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Mo... ...became the father of graph theory. In 1847 Kirchoff developed the theory of trees, in order to solve the system of simultaneous linear equations, wh...

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...In The 2nd Conference on Combinatorics and Graph Theory of China (Aug. 16-19, 2006, Tianjing), I formally presented a combinatorial conjecture on mathematical sciences (abbreviated to CC Conjecture), i.e., a mathematical science can be reconstructed from or made by co...

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...Contents Preface to the Second Edition . . . . . . . . . . . . . . . . . . . i Chapter 1. Combinatorial Principle with Graphs . . . . . . . . . . 1 1.1 Multi-sets with operations. . . . . . . . . . . . . . . . . . . . .2 1.1.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Operation . . . . . . . . . . . . . . . . . . . . . ...

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...OF SOCIAL ASPECTS OF MIGRANT LABOURERS LIVING WITH HIV/AIDS USING FUZZY THEORY AND NEUTROSOPHIC COGNITIVE MAPS With Specific Reference To Rura... ... OF SOCIAL ASPECTS OF MIGRANT LABOURERS LIVING WITH HIV/AIDS USING FUZZY THEORY AND NEUTROSOPHIC COGNITIVE MAPS With Specific Reference To Rura... ... are used in the study of the problem. In chapter III yet another fuzzy theory technique namely the bi-directional associative memories is used in... ...and made on the mathematical knowledge of the reader is very meager. Also graphs are given to make the understanding simple and easy. The tools used... ...0 0 1 0 The row sum matrix ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − 6 4 1 20 The graph depicting the maximum age group of the migrant labourers who are aff... ...ix Age Group 21 The row sum matrix ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − 6 5 1 The graph depicting the maximum age group of the migrant labourers who are aff... ...Ws. He explains when the men went on trips, the CSWs would be under the trees to offer their services. He also adds that the CSWs are of three typ... ...mployed as ‘Saanars’ where their work was to clean and manure the coconut trees and pluck coconuts and they complained of no work for in several pl... ...coconuts and they complained of no work for in several places the coconut trees had dried or did not give any yield. d. The other migrant laboure...

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...ose of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical a...

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...Sequences on Graphs with Symmetries Linfan Mao Chinese Academy of Mathematics and System Science, Beijing, 10080, P.R.China Beijing Institute of Civil Engineering and Architecture, Beijing, 100044, P.R.China E-mail: maolinfan@163.com ...

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...Contents Lucas Graceful Labeling for Some Graphs BY M.A.PERUMAL, S.NAVANEETHAKRISHNAN AND A.NAGARAJAN . . . . . 01 Sequences on Graphs with Symmetries BY LINFAN MAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Supermagic Coverings of Som...

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...ose of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical a...

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...opological Vector Spaces BY SAYED ELAGAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Path Double Covering Number of Product Graphs BY A. NAGARAJAN, V. MAHESWARI, S. NAVANEETHAKRISHNAN. . . . . . 27 Some Remarks on Fuzzy N-Normed Spaces BY SAYED ELAGAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 σ-Coloring of the Monohed...

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...A combinatorial map is a connected topological graph cellularly embedded in a surface. As a linking of combinatorial configuration with the classical mathematics, it fascinates more and more mathematician’s interesting. Its function and role in mathematics are widely acce...

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... States of America Preface A combinatorial map is a connected topological graph cellularly embedded in a surface. As a linking of combinatorial config... ...of those is surveyed in Liu’s book [33]. To deter- mine the embedding of a graph on surfaces, including coloring a map on surfaces is another trend in... ... on surfaces, including coloring a map on surfaces is another trend in map theory. Its object is combinatorialization of surfaces, see Gross and Tucke... ...on surfaces, related maps with groups and geometry is a glimmer of the map theory with other mathematics. In fact, maps as a kind of the decomposition... ...sition of surfaces, should be given more attention to its role in surfaces theory, such as the Riemann surfaces, Klein surfaces and manifolds theory. ... ...ces, Smarandache geometries, maps and the semi-arc automorphism group of a graph. Preface ii A relation for maps and Smarandache manifolds (abbreviat... ... propertyP tobeitsakind ofsubgraphs, such as, regular subgraphs, circuits, trees, stars, wheels, ···. Let M = (X α,β ,P) be Chapter 2 On the Automorp... ...he greatest common divisor of a,b,···. Corollary 2.3.2 Let T be the set of trees in the map M. Then for ∀G AutM, |G| | (2lt l ,l≥1), where t l denotes... ...p M. Then for ∀G AutM, |G| | (2lt l ,l≥1), where t l denotes the number of trees with l edges. Corollary 2.3.3 Let v i be the number of vertices with ...

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...ose of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical a...

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...ive Generalized Disjunctive Fuzzy Nonlinear Fractional Programming BY E.E.AMMAR . . . . . . . . . . . . . . . . . . . . 01 Surface Embeddability of Graphs via Joint Trees BY YANPEI LIU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 Plick Graphs with Crossing Number 1 BY B.BASAVANAGOUD AND V.R.KULLI . . . . . . . . . . . . . . . . . . . . . . ...

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...se of algebraic multi-systems, multi-metric spaces,· · · , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical ...

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...Halfsubgroups BY ARUN S. MUKTIBODH. . . . . . . . . . . . . . . . . . . . . . . . . . . . .01 Flexibility of Embeddings of a Halin Graph on the Projective Plane BY HAN REN and YUN BAI. . . . . . . . . . . . . . . . . . . . . . . . . . . .06 Curvature Equations on Combinatorial Manifolds with Applications to Theoretical Physics BY LINFAN MAO. . . . ....

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... untouchability and the caste-system of Hinduism using Fuzzy and Neutrosophic theory. In studying untouchability, we have chosen to view it throug... ...te is based on the 8 mind, it is appropriate to use Fuzzy and Neutrosophic theory. In Chapter Two we use the opinion of several experts to anal... ...re, we use the tools that have been described in Chapter One. Fuzzy Directed Graphs and Neutrosophic Directed Graphs of these Fuzzy and Neutrosophi... ... and Neutrosophic models happen to be very dense. We have represented 16 such graphs in this book. In Chapter Three we give a brief introduction ab... ...od is most simple and an effective one as it can analyse the data by directed graphs and connection matrices and give the hidden pattern of the syst... ...that gives the hidden pattern of the situation. As we have a very well known theory, which states that the strength of the data depends on, the num... ...Manu. Further these people should live unseen, away from the city, under the trees or near the graveyards and tend dogs and donkeys and should not ...

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.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 D. Rabounski A Theory of Gravity Like Electrodynamics. . . . . . . . . . . . . . . . . . .... ...sova Gravitational Waves and Gravitational Inertial Waves in the Gen- eral Theory of Relativity: A Theory and Experiments. . . . . . . . . . . . . . .... ...i, L. Borissova, F. Smarandache Quantum Causality Threshold in the General Theory of Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . .... ...ce frame are chronometrically invariant quantities (see the previous para- graph for the details). Keeping this fact in mind, Zelmanov formulated his ... ...a later moment. Based on this physical picture of the electron self-energy graph, we will evaluate the Geometric Probability that an electron emits a ... ...tails about Shilov boundaries, the conformal group, future tubes and holo- graphy we refer to the article by Gibbons [12] and [7, 16]. In order to def... ...nce distributions from monadi. Now consider the number of different N-node trees, with the same distance distribution {D k }, to which i can belong. B...

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...thus opening new ways of research in four fields: philosophy, logics, set theory, and probability/statistics. It was known to me his setting up in 1... ...so, to experiment." 2.2. Etymology: Paradoxism = paradox+ism, means the theory and school of using paradoxes in literary, artistic, philosophical... ...thiness, triteness, and non­ poeticality; - paralinguistic verse (only!): graphics, lyrical portraits, drawings, drafts... - non-words and non-sente... ...th-West of the United States by a single person!); - super-poems; 6 - graphic poems; - upset-poems.] 3. “Dictionary of Computing”, by Denis How... ... with different ideational spectra. Neutrosophy considers a proposition, theory, event, concept, or entity, "A" in relation to its opposite, "Anti- ... ...g towards a mathematization of philosophy, but not in a Platonian sense. Graph 5. O.a.1: Materialism % Idealism = constant, for any society. S P ... ...] Dunn, J. M., Intuitive Semantics for First Degree Entailment and Coupled Trees, Philosophical Studies, Vol. XXIX, pp. 149-68, 1976. [59] Eksioglu,...

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...ening address. "Fellow students: members of the faculty, hiding behind the trees; fair sisters of North Adams; members of the A. P. W. society, the ad... ...y .lollinni in cliiipter !t, verses fS 12, in the Hook of .liiilges. iMimy trees wiuited a louder ant! the erown was olTored lotholig, the olive and t... ...cles there is a har- vest of work for those v.ho carry into practice their theory that Williamstown can be beautified. The nest issue of The Record wi... ...ss. SEPT, 30, 1907 —Reoord oandidates meet in 8 J. H. tonight at 7.30. The Theory of Grievance Against Customs Abolition By the widely varying view po... ...ognition as a college interest. If only as an element of dis- proof to the theory maintained by some that intercollegiate interest in basketball is dy... ...en acre farm. 50 acres tillable, fiO acres pasture and woodland, 200 sugar trees, cuts 30 tons hay, one and one-half story house. 3li x 40; 28 s 40 ba... ...the quartet, six men strong. The Derby, not a hat. but a kingly sport, was graphically represented by fractious wooden horses, astride of which were e... ...d meet bis classes in Theism within a few days. Qrinds, drawing and photo- graphs of local interest are desired for the 1909 Gul. Drawings should be s... ...Beatrice Her- ford. Photochrome Views Exhibited An exhibition of 47 photo- graphs has been placed on the art racks in tho second floor corri- dor of H...

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...ography IV. Appendix – The Keynesian view of Unemployment, Stabilization Theory and Full Employment Budgeting V. Appendix – Unemployment throughout... ...le nature conservation work for the young and unmarried men. They planted trees, erected flood barriers, put out forest fires and constructed forest... ...uctivity, Comparative advantage and Economic Growth”, Journal of Economic Theory 58, 1992 17. “Deindustrialization”, World Economic Outlook, April 1... ... Appendix – The Keynesian view of Unemployment, Stabilization Theory and Full Employment Budgeting (From The Encyclopedia Britannica 1... ...he USA in 1932. It offered work for young and unmarried men. They planted trees, erected flood barriers, put out forest fires, and constructed fores... ...as more people retire and others migrate. At a certain point in time, the graphs diverge, leaving in their wake disgruntled and cheated pensioners a...

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...nly in 1998, did the Britannica switch to multimedia and added tables and graphs to the CD. Video and sound were to make their appearance even later.... ...nd then in color), tables, detailed maps and anatomical charts, and other graphics to their books. Battles fought between publishers-librarians over ... ...is case be more readily tolerated as a marginal phenomenon. This is the theory. But the facts are tellingly different. The less the cost of product... ...not be bought. We will adapt by doing the entire public domain, including graphics, music, movies, sculpture, paintings, photographs, etc. . . . Q.... ...hington state, there were plenty of wild blackberries, raspberries, apple trees, pear trees, plum trees, grapes. I never even considered buying any o... ...s overhead to get their ever- decreasing catalog of items, printed on dead trees, delivered to shopping malls. Fear of illicit copying (music and v... ...is not the main function of the advertising dollar. Modern economic signal theory has cast advertising in a new and surprising - though by no means c... ...prising - though by no means counterintuitive - light. According to this theory, the role of advertising is to signal to the marketplace the adverti... ...memory of his search engine and assist it in designing its decision making trees. The engine will become an electronic friend, advise the user, even ...

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...he early part of that period of the present century which we now call ?Empire.? Rain had refreshed the earth during the month of October, so that the trees were still green and leafy in November. The French people were beginning to put faith in a secret understanding between the skies and Bonaparte, then declared Consul for life,--a belief in which that man owes part of hi...

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...re.” Rain had refreshed the earth during the month of October, so that the trees were still green and leafy in November. The French people were beginn... ...cted by a railing, beyond which is a wide and deep ha-ha, full of vigorous trees, its parapets bristling with iron arabesques, the innumerable sharp p... ..., within the park, a corresponding half-circle is formed by groups of rare trees. The pavilion, therefore, stands at the centre of this round open spa... ...sion of the social sys- tem. In these days we are so far removed from this theory of primitive people that almost all women, even those who do not des... ...nd was able to ap- ply them in this affair with terrible celerity, under a theory, now converted into a certainty, of the guilt of Michu and the Messi... ...t, who was unable at that late hour, it being after dark, to use the tele- graph. They therefore sent a mounted messenger to Paris to notify the minis...

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...s. There was a Boy once who had been brought up under the “sheltered life” theory; and the theory killed him dead. He stayed with his people all his d... ...was the bed- ding, and on the table lay The Boy’s writing-case with photo- graphs. He had gone away to die like a poisoned rat! The Major said to hims... ... side. Strickland had himself to thank for this. He held the extraordinary theory that a Policeman in India should try to know as much about the nativ... ...olish enough to take that man for his model; and, following out his absurd theory, dabbled in un- savory places no respectable man would think of expl... ...are very few indeed. Then we talked in groups or together, lying under the trees, with the sun-baked roses dropping their petals on our feet, until su... ...it until the moon went out and a burning hot wind began lashing the orange-trees with a sound like the noise of the sea. Before we knew where we were,... ... the Hills fear it might be blown down. So we felt our way to the or- ange-trees where the horses were picketed and waited for the storm to blow over.... ...ents, and fi- nally communicates, say, with a friend of yours in the Tele- graph, who once wrote some notes on the customs of the gold-washers when he...

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...was sitting out here in front of my cabin, with my cat, taking the sun, and looking at the blue hills, and listening to the leaves rustling so lonely ... ...ell over backward suffocating with laughter, and the next jay took his place and done the same. “Well, sir, they roosted around here on the housetop a... ...iddle of the sixteenth century, while the 51 A Tramp Abroad Saviour was nude, with the exception of a cloth around the loins. We had dinner under the... ... been boiled. The Germans are pretty sure to boil a trout or anything else if left to their own devices. This is an argument of some value in support ... ...s and paying costs of rehabilitation. We had hopes of finding the owner there, distributed around amongst the rocks, for it would have made an elegant... ...rs object that water boils at a lower and lower tem- perature the higher and higher you go, and hence the appar- ent anomaly. I answer that I do not b... ...aratus at it without the loss of an instant, and should have got an elegant picture if my don- key had not interfered. It was my purpose to draw this ... ...ut with the ice-axes, and it proved to be a piece of the undressed skin of some animal—a hair trunk, perhaps; but a close inspection disabled the hair... ...water! The denudation of the land was upon a grand scale. All superficial accumulations were swept away, and the bed- rock was exposed. It was describ...