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..., Mohar and Thomassen [53] and White [70], especially the [53] for detail. The construction of regular maps on surfaces, related maps with groups and geometry is a glimmer of the map theory with other mathematics....

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...D SMARANDACHE GEOMETRIES American Research Press Rehoboth, NM 2005 Linfan MAO Institute of Systems Science Academy of Mathematics and Systems Chinese... ...sfield, PA 16933, USA. Copyright 2005 by American Research Press and Linfan Mao Rehoboth, Box 141 NM 87322, USA Many books can be downloaded from the f... ...The construction of regular maps on surfaces, related maps with groups and geometry is a glimmer of the map theory with other mathematics. In fact, ma... ...zation or making combinatorial refinement for the Riemann surfaces, Riemann geometry and finally, the Smarandache geometries by maps is a very interesti... ...ntribution of combinatorics to classical mathematics, such as, the Riemann geometry and Smarandache geometries is realized. Those are the wish of mine... ...l presented here. Any suggestions for improving this book are welcome. L.F.Mao AMSS, Beijing June, 2005 Contents Preface................................ ...Each of the later two is proposed by denial the 5th postulate for parallel lines in the Euclid postu- lates of geometry. The Smarandache geometries is... ...the embedding sequence of the graph Γ. Define a functionF(x ¯ ,y ¯ ) to bey-linear if it can be represented as the following form F(x ¯ ,y ¯ ) =f(x 1 ,... ...x 2 ,···) Λ∈O Λ(y ¯ ), where, I denotes a subset of index and O a set of linear operators. Notice that f(x 1 ,x 2 ,···) =F(x ¯ ,0 ¯ ), where 0 ¯ = (...

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...nal research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,...

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... . 22 Special Smarandache Curves in the Euclidean Space BY AHMAD T. ALI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 The H-Line Signed Graph of a Signed Graph BY R.RANGARAJAN, M. S. SUBRAMANYA AND P. SIVA KOTA REDDY. . . .37 Min-Max Dom-Saturation Number of a Tree BY S. ARUMUGAM AND S. SUDHA . . . . . . . . . . . . . . . . . . . . . . . 45 Em...

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...nal research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,...

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...s On (∈ vq)- Fuzzy Bigroup BY AKINOLA L.S. and AGBOOLA A.A.A. . . . . . . . . . . . . . . . . . . . . . . . . . . 01 Connectivity of Smarandachely Line Splitting Graphs BY B.BASAVANAGOUD and VEENA MATHAD . . . . . . . . . . . . . . . . . . . . . . 08 Separation for Triple-Harmonic Di?erential Operator in Hilbert Space BY E.M.E.ZAYED and S.A.OMRAN . . . . . . . . . . ...

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...nal research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,...

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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 Abstract: An interestin...

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... 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 Some Simple Graphs BY P.JEYANTHI AND P.SELVAGOPAL . . . . . . . . . . . . . . . . . . . . . . . 33 Elementary Abelia...

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...nal research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,...

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... . . . 85 Weak and Strong Reinforcement Number For a Graph BY PINAR DUNDAR, TUFAN TURACI and DERYA DOGAN. . . . . . . . . . . . . . 91 Tulgeity of Line, Middle and Total Graph of Wheel Graph Families BY AKBAR ALI.M.M, S.PANAYAPPAN and VERNOLD VIVIN.J . . . . . . . . . . . . 98 Labeling, Covering and Decomposing of Graphs – Smarandache’s Notion in Graph Theory BY LINF...