MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES), Volume 1 / 2015

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Papers on Antidegree Equitable Sets in a Graph, One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs, A New Approach to Natural Lift Curves of the Spherical Indicatrices of Timelike Bertrand Mate, On Signed Graphs Whose Two Path Signed Graphs are Switching Equivalent to Their Jump Signed Graphs, and other topics.Contributors: C. Adiga, K.N.S. Krishna, Mathew Varkey T.K, Sunoj B.S, V. Ramachandran, C. Sekar, W. Barbara, P. Sugirtha, R. Vasuki, J. Venkateswari, Yizhi Chen, Siyan Li, Wei Chen, and others.

Text of MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES), Volume 1 / 2015

  • ISBN 978-1-59973-337-1

    VOLUME 1, 2015

    MATHEMATICAL COMBINATORICS

    (INTERNATIONAL BOOK SERIES)

    Edited By Linfan MAO

    THE MADIS OF CHINESE ACADEMY OF SCIENCES AND

    ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS

    March, 2015

  • Vol.1, 2015 ISBN 978-1-59973-337-1

    MATHEMATICAL COMBINATORICS

    (INTERNATIONAL BOOK SERIES)

    Edited By Linfan MAO

    The Madis of Chinese Academy of Sciences and

    Academy of Mathematical Combinatorics & Applications

    March, 2015

  • Aims and Scope: The Mathematical Combinatorics (International Book Series) is a

    fully refereed international book series, quarterly comprising 100-150 pages approx. per volume,

    which publishes original 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, , etc.. Smarandache geometries;Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and

    map enumeration; Combinatorial designs; Combinatorial enumeration;

    Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential

    Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations

    with Manifold Topology;

    Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi-

    natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields;

    Mathematical theory on parallel universes; Other applications of Smarandache multi-space and

    combinatorics.

    Generally, papers on mathematics with its applications not including in above topics are

    also welcome.

    It is also available from the below international databases:

    Serials Group/Editorial Department of EBSCO Publishing

    10 Estes St. Ipswich, MA 01938-2106, USA

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    and

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    Indexing and Reviews: Mathematical Reviews (USA), Zentralblatt Math (Germany), Refer-

    ativnyi Zhurnal (Russia), Mathematika (Russia), Directory of Open Access (DoAJ), Interna-

    tional Statistical Institute (ISI), International Scientific Indexing (ISI, impact factor 1.416),

    Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA).

    Subscription A subscription can be ordered by an email directly to

    Linfan Mao

    The Editor-in-Chief of International Journal of Mathematical Combinatorics

    Chinese Academy of Mathematics and System Science

    Beijing, 100190, P.R.China

    Email: maolinfan@163.com

    Price: US$48.00

  • Editorial Board (3nd)

    Editor-in-Chief

    Linfan MAO

    Chinese Academy of Mathematics and System

    Science, P.R.China

    and

    Academy of Mathematical Combinatorics &

    Applications, USA

    Email: maolinfan@163.com

    Deputy Editor-in-Chief

    Guohua Song

    Beijing University of Civil Engineering and

    Architecture, P.R.China

    Email: songguohua@bucea.edu.cn

    Editors

    S.Bhattacharya

    Deakin University

    Geelong Campus at Waurn Ponds

    Australia

    Email: Sukanto.Bhattacharya@Deakin.edu.au

    Said Broumi

    Hassan II University Mohammedia

    Hay El Baraka Ben Msik Casablanca

    B.P.7951 Morocco

    Junliang Cai

    Beijing Normal University, P.R.China

    Email: caijunliang@bnu.edu.cn

    Yanxun Chang

    Beijing Jiaotong University, P.R.China

    Email: yxchang@center.njtu.edu.cn

    Jingan Cui

    Beijing University of Civil Engineering and

    Architecture, P.R.China

    Email: cuijingan@bucea.edu.cn

    Shaofei Du

    Capital Normal University, P.R.China

    Email: dushf@mail.cnu.edu.cn

    Baizhou He

    Beijing University of Civil Engineering and

    Architecture, P.R.China

    Email: hebaizhou@bucea.edu.cn

    Xiaodong Hu

    Chinese Academy of Mathematics and System

    Science, P.R.China

    Email: xdhu@amss.ac.cn

    Yuanqiu Huang

    Hunan Normal University, P.R.China

    Email: hyqq@public.cs.hn.cn

    H.Iseri

    Mansfield University, USA

    Email: hiseri@mnsfld.edu

    Xueliang Li

    Nankai University, P.R.China

    Email: lxl@nankai.edu.cn

    Guodong Liu

    Huizhou University

    Email: lgd@hzu.edu.cn

    W.B.Vasantha Kandasamy

    Indian Institute of Technology, India

    Email: vasantha@iitm.ac.in

    Ion Patrascu

    Fratii Buzesti National College

    Craiova Romania

    Han Ren

    East China Normal University, P.R.China

    Email: hren@math.ecnu.edu.cn

    Ovidiu-Ilie Sandru

    Politechnica University of Bucharest

    Romania

  • ii International Journal of Mathematical Combinatorics

    Mingyao Xu

    Peking University, P.R.China

    Email: xumy@math.pku.edu.cn

    Guiying Yan

    Chinese Academy of Mathematics and System

    Science, P.R.China

    Email: yanguiying@yahoo.com

    Y. Zhang

    Department of Computer Science

    Georgia State University, Atlanta, USA

    Famous Words:

    Nothing in life is to be feared. It is only to be understood.

    By Marie Curie, a Polish and naturalized-French physicist and chemist.

  • Math.Combin.Book Ser. Vol.1(2015), 1-13

    NC Smarandache Curves of Mannheim Curve CoupleAccording to Frenet Frame

    Suleyman SENYURT and Abdussamet CALISKAN

    (Faculty of Arts and Sciences, Department of Mathematics, Ordu University, 52100, Ordu/Turkey)

    E-mail: senyurtsuleyman@hotmail.com

    Abstract: In this paper, when the unit Darboux vector of the partner curve of Mannheim

    curve are taken as the position vectors, the curvature and the torsion of Smarandache curve

    are calculated. These values are expressed depending upon the Mannheim curve. Besides,

    we illustrate example of our main results.

    Key Words: Mannheim curve, Mannheim partner curve, Smarandache Curves, Frenet

    invariants.

    AMS(2010): 53A04

    1. Introduction

    A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame

    vectors on another regular curve, is called a Smarandache curve ([12]). Special Smarandache

    curves have been studied by some authors .

    Melih Turgut and Suha Ylmaz studied a special case of such curves and called it Smaran-

    dache TB2 curves in the spaceE41 ([12]). Ahmad T.Ali studied some special Smarandache curves

    in the Euclidean space. He studied Frenet-Serret invariants of a special case ([1]). Muhammed

    Cetin , Ylmaz Tuncer and Kemal Karacan investigated special Smarandache curves according

    to Bishop frame in Euclidean 3-Space and they gave some differential goematric properties of

    Smarandache curves, also they found the centers of the osculating spheres and curvature spheres

    of Smarandache curves ([5]). Senyurt and Calskan investigated special Smarandache curves in

    terms of Sabban frame of spherical indicatrix curves and they gave some characterization of

    Smarandache curves ([4]). Ozcan Bektas and Salim Yuce studied some special Smarandache

    curves according to Darboux Frame in E3 ([2]). Nurten Bayrak, Ozcan Bektas and Salim Yuce

    studied some special Smarandache curves in E31 [3]. Kemal Tas.kopru, Murat Tosun studied

    special Smarandache curves according to Sabban frame on S2 ([11]).

    In this paper, special Smarandache curve belonging to Mannheim partner curve suchas NC drawn by Frenet frame are defined and some related results are given.

    1Received September 8, 2014, Accepted February 12, 2015.

  • 2 Suleyman SENYURT and Abdussamet CALISKAN

    2. Preliminaries

    The Euclidean 3-space E3 be inner product given by

    , = x21 + x32 + x23

    where (x1, x2, x3) E3. Let : I E3 be a unit speed curve denote by {T,N,B} themoving Frenet frame . For an arbitrary curve E3, with first and second curvature, and respectively, the Frenet formulae is given by ([6], [9])

    T = N

    N = T + BB = N.

    (2.1)

    For any unit speed : I E3, the vector W is called Darboux vector defined by

    W = (s)T (s) + (s) +B(s).

    If consider the normalization of the Darboux C =1

    WW , we have

    cos =(s)

    W , sin =(s)

    W ,

    C = sinT (s) + cosB(s) (2.2)

    where (W,B) = . Let : I E3 and : I E3 be the C2 class differentiable unitspeed two curves and let {T (s), N(s), B(s)} and {T (s), N(s), B(s)} be the Frenet frames ofthe curves and , respectively. If the principal normal vector N of the curve is linearlydependent on the binormal vector B of the curve , then () is called a Mannheim curve and() a Mannheim partner curve of (). The pair (, ) is said to be Mannheim pair ([7], [8]).The relations between the Frenet frames {T (s), N(s), B(s)} and {T (s), N(s), B(s)} are asfollows:

    T = cos T sin BN = sin T + cos B

    B = N

    (2.3)

    cos =

    dsds

    sin = dsds

    .(2.4)

    where (T, T ) = ([8]).

    Theorem 2.1([7]) The distance between corresponding points of the Mannheim partner curves

    in E3 is constant.

  • NC Smarandache Curves of Mannheim Curve Couple According to Frenet Frame 3

    Theorem 2.2 Let (, ) be a Mannheim pair curves in E3. For the curvature