Volume-6 , Issue-3 , Jun 2019, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  Weakly Essential Supplemented Modules

  Sushma Jat and Vivek Prasad Patel

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.79-83, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.7983

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  Abstract
  In this work we extend supplemented modules as weakly essential supplemented modules and some properties of these modules are investigated. We show that every finite sum of weakly essential supplemented modules is weakly essential supplemented

  Key-Words / Index Term
  Essential Submodules, Small Submodules, Supplemented Modules, 2010 Mathematics Subject Classification: 16D10, 16D70

  References
  [1]. Alizade, R.; B¨uy¨ukas¸ik, E. – Cofinitely weak supplemented modules, Comm. Algebra, 31 (2003), 5377–5390.
  [2]. Alizade, R.; B¨uy¨ukas¸ik, E. – Extensions of weakly supplemented modules, Math. Scand., 103 (2008), 161–168.
  [3]. B¨uy¨ukas¸ik, E.; T¨urkmen, E. – Strongly radical supplemented modules, Ukrainian Math. J., 63 (2012), 1306-1313.
  [4]. B¨uy¨ukas¸ik, E.; Lomp, C. – Rings whose modules are weakly supplemented are perfect. Applications to certain ring extensions, Math. Scand., 105 (2009), 25–30.
  [5]. Clark, J.; Lomp, C.; Vanaja, N.; Wisbauer, R. – Lifting Modules, Supplements and projectivity in module theory. Frontiers in Mathematics. Birkh¨auser Verlag, Basel, 2006.
  

  Citation
  Sushma Jat and Vivek Prasad Patel, "Weakly Essential Supplemented Modules," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.79-83, 2019

• Open Access   Article

  Multi Attribute Decision-Making Method Using Value and Ambiguity in Single Valued Hexagonal Neutrosophic Numbers

  A.SahayaSudha and K.R. Vijayalakshmi

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.84-90, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.8490

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  Neutrosophic number is a new mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. The aim of this paper is to develop a method for ranking single valued hexagonal neutrosophic numbers (SVHNNs) in the process of decision making in the neutrosophic environment.As a part of it a single valued Hexagonal neutrosophic number is developed in the multi criteria frame work. The concept of SVHNNs is introduced; Arithmetic operations and cut sets over SVHNNs are investigated. Then, the values and ambiguities of the Truth membership degree indeterminacy and false membership degree for SVHNNs are defined with value-index and ambiguity-index. Finally, a value and ambiguity-based ranking method is developed and applied to solve multi-attribute decision making problems in which the ratings of alternatives on attributes are expressed using SVHNNs. The proposed framework consists of four activities, selecting the problem,gathering information, computing and ranking the alternatives.

  Key-Words / Index Term
  Single valued hexagonal neutrosophic numbers, Value index, Ambiguity index, MADM strategy

  References
  [1] I.Deli and Y.Subas “A ranking method for single valued neutrosophic numbers and its applications to multi-attribute decision making problems” .International journal of Machine learning and cybernetics. Vol. 8 Issue.4 pp.1309–1322, 2016
  [2] I.Deli and Y.Subas “Single valued neutrosophic numbers and their applications to multi-criteria decision making problem”. Neutrosophic Sets and Systems Vol. 2 No.1, pp.1–13, 2014
  
  [3] J.Chen,J.Ye” Some single-valued Neutrosophic Dombi weighted Aggregation operators for multiple attribute decision making”. Symmetry,Vol.9 Issue.6, pp.82-92, 2017
  
  [4] J. Peng,J.Wang,X.Wu & X.Chen “ Multi-valued neutrosophic sets and power aggregation operators with their applications in multi-criteria group decision-making problems”. International Journal of Computation Systems, Vol.8,Issue.2 pp.345-363,2014.
  
  [5]F. Smarandache, “A unifying field in logics. Neutrosophy: Neutrosophic probability”, American Research Press, Rehoboth ISBN 978-1-59973-080-6, 1998
  [6]F.Smarandache “Neutrosophic set a generalization of intuitionistic fuzzy set”. International Journal of Pure and Applied Mathematics, Vol.24,Issue .3, pp.287-297,2005
  [7] H.Wang, F.Smarandache and R.Sunderraman. “Single valued neutrosophic sets”, Technical Science and Applied Mathematics Vol.4, pp.410-413, 2010
  [8] H.Wang,F.Smarandache, and YQ.Zhang, “Interval neutrosophic sets and logic”: Theory and applications in computing, Hexis,Neutrosophic book series Book 5 (2005)
  
  [9]J.Ye “Multi-criteria decision-making method using the correlation coefficient under single-valued neutrosophic environment”. International Journal of general systems Vol.42,Issue.4, pp.386-394,2013
  [10] J.Ye, “A multi-criteria decision making method using aggregation operators for simplified neutrosophic sets”. International Journal of Fuzzy Sets and Systems. Vol.26,Issue.5, pp.2459-2466,2014.

  Citation
  A.SahayaSudha and K.R. Vijayalakshmi, "Multi Attribute Decision-Making Method Using Value and Ambiguity in Single Valued Hexagonal Neutrosophic Numbers," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.84-90, 2019

• Open Access   Article

  A Mathematical Model on Chain Of Competitions in Different Trophic Levels

  Nabajyoti Bhattacharjee

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.91-93, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.9193

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  Abstract
  In this paper we shall consider a sequence of trophic levels and obtain a recurrence for the total biomass production in each trophic level consisting of exactly two species competing for resource in its immediate lower trophic level and prove that if the consumption factor of both the species in each trophic level is same and the consumption factor at each trophic level forms a strictly decreasing sequence and all are greater than one, then biomass production at each trophic level forms a strictly decreasing sequence of total biomass production starting from lowest trophic level.

  Key-Words / Index Term
  Mathematical modelling, Competition, Trophic level, Superiority factor, chain of competition

  References
  [1]. N. Bhattacharjee, Bio-Mathematical Model on Competition between Two Species, IJMAA, Vol. 5, 3-A(2017), pp-69-83, 2017.
  
  [2]. J. H. Connell, The influence of interspecific competition and other factors on the distribution of the barnacle Chthamalus stellatus, JSTOR, Vol. 42, No. 4, 1961.
  
  [3]. P. L. Munday, G. P. Jones, M. J. Caley, Interspecific Competition and Coexistence in a guild of Coral-Dwelling Fishes, Ecological Society of America, 82(8), pp-2177-2189, 2001.
  
  [4]. D. E. Goldberg, A. M. Barton, Patterns and Consequences of Interspecific Competition in Natural Communities: A Review of Field Experiments with Plants , The American Naturalist, 139(4), pp-771-801, 1992
  
  [5]. Lindeman, RL, The trophic-dynamic aspect of ecology, Ecology, 23, pp-399-418, 1942
  

  Citation
  Nabajyoti Bhattacharjee, "A Mathematical Model on Chain Of Competitions in Different Trophic Levels," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.91-93, 2019

• Open Access   Article

  Separation Axioms In Topological Spaces Via Ideal

  J. K. Maitra and Rajesh Kumar Tiwari

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.94-96, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.9496

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  Abstract
  In this article, D^⋆ sets used to characterize some weak separation axioms in ideal topological spaces and to study some of their essential properties.

  Key-Words / Index Term
  Ideal topological spaces, Open set

  References
  [1]. R. Engelking, “General Topology”, second edition, Sigma Series in Pure Mathematics 6, HeldermannVerlag, Berlin, 1989.
  [2]. D. Jankovic and T. R. Hamlet, “New topologies from Old via Ideals”, The American Mathematics Monthly, Vol. 97, Issue. 4, pp. 295-310, 1990. Transaction, Vol.31, Issue.4, pp.123-141, 2012.
  [3]. M. S. Sarsak, Weak Separation axioms in Generalized Topological Space, Acta Math. Hungar, 131 (1-2),110-121, 2010.
  [4]. Rajesh Kumar Tiwari1, J. K. Maitra and R. Vishwakarma, A Generalized form of G^*-open Sets, International Journal of Scientific Research Mathematical and Statistical Science, Vol. 5, issue. 6, pp. 53-58, 2018.
  

  Citation
  J. K. Maitra and Rajesh Kumar Tiwari, "Separation Axioms In Topological Spaces Via Ideal," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.94-96, 2019

• Open Access   Article

  Verification of Birthday Paradox

  Yogesh R Yewale

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.97-99, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.9799

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  Birthday problem is one of the famous problem in the probability theory. It is also known as birthday paradox in some literature. In this paper attempt has been made to discuss the verification of the birthday paradox by collecting real life data. Birthdates of one thousand individuals are considered for this study. These individuals are divided into the various groups of size n. In each group matching pairs of birthdates were observed and the number of groups with matched pairs of birthdates were counted. This procedure is then repeated for different values of group size n. Chi-square test of goodness of fit is used to decide whether the discrepancies between the observed and expected values of the number of groups with matched birthdates is insignificant or not. It is found that the discrepancies between observed and expected values of the number of matched pairs is insignificant at 5% level of significance, which verifies the statement of the birthday paradox.

  Key-Words / Index Term
  Birthday paradox, Chisquare test of goodness of fit

  References
  [1] J. M. Johnson, “THE BIRTHDAY PROBLEM EXPLAINED,” Math. Teach., vol. 90, no. 1, pp. 20–22, 1997.
  [2] K. R. Rebman, “The Pigeonhole Principle ( What It Is , How It Works , and How It Applies to Map Coloring ),” Two-Year Coll. Math. J., vol. 10, no. 1, pp. 3–13, 1979.
  [3] J. D. Austin, “The birthday problem revisited,” Two-Year Coll. Math. J., vol. 7, no. 4, pp. 39–42, 1976.
  [4] K. S. Jones, “The Birthday Problem Again ?,” Math. Teach., vol. 86, no. 5, pp. 373–377, 1993.

  Citation
  Yogesh R Yewale, "Verification of Birthday Paradox," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.97-99, 2019

• Open Access   Article

  An Analysis into the Efficiency of Methods to Solve Flpp

  Alka Benny and Megha Sukumaran

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.100-104, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.100104

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  Abstract
  In this paper, we first change the fuzzy linear programming problem into the fuzzy linear system of equations. Then a Fourier Motzkin elimination and an interior point technique methods are discussed to solve the above converted fuzzy linear programming problem to find an approximate optimal solution. A comparison is made with the above two methods with other two methods of LPP.

  Key-Words / Index Term
  fuzzy variable linear programming, triangular fuzzy number, interior point method, fuzzy set, fuzzy linear system of equations, LPP, fuzzy LPP, graphical method.

  References
  [1] A. Nagoor Gani and S. N. Mohamed Assarudeen, “An algorithmic approach of solving linear system using Fourier Motzkin elimination method”, Advances in Fuzzy Sets and Systems, Vol.10, Number 2,2011
  [2] A. Nagoor Gani and S. N. Mohamed Assarudeen, “An approximate optimal solution for the Fuzzy Variable Linear Programming Problem using Interior Point Technique”, International Journal of Pure and Applied Mathematics, Vol.85 No.2 2013, 395-404, ISSN: 1311-8080.
  [3] S. H. Chen and C. H. Hsieh, Graded mean integration representation of generalized fuzzy number. Chinese Fuzzy System Assoc. 5(2) (1999), 1-7.
  [4]M. Friedman, M. Ming and A. Kandel, Fuzzy Sets and Systems,201-209. 1998
  [5]Alexander Schjver, Theory of Linear and Integer Programming. John Wiley and Sons Ltd., 1998.

  Citation
  Alka Benny and Megha Sukumaran, "An Analysis into the Efficiency of Methods to Solve Flpp," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.100-104, 2019

• Open Access   Article

  Fixed Point Theorem of Generalized Contraction Function in Fuzzy Symmetric Spaces

  R. Krishnakumar and Nagaral Pandit Sanatammappa

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.105-110, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.105110

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  Abstract
  In this paper the existences of the unique fixed point of a generalised contraction function have been studied in fuzzy symmetric spaces instead of metric space. And we have generalized the fixed point theorems for complete self-mapping in fuzzy symmetric spaces. These results improve and generalize some important known results in existing literature.

  Key-Words / Index Term
  Fixed point, Fuzzy symmetric spaces, contraction function, t-norm, non-decreasing

  References
  [1] M. Abbas and B. E. Rhoades, “Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized condition of integral type”, Fixed Point Theory Appl., (2007), Art. ID 54101, 9pp.
  [2] H. Aydi, E. Karapinar, and S. Moradi, “Coincidence points for expansive mappings under c-distance in
  cone metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2012,Article
  ID 308921, 2012.
  [3] A. George, P. Veeramani, On Some result in fuzzy metric spaces, Fuzzy Sets and Systems, Vol.64,
  pp.395-399,1994.
  [4] A. George, P. Veeramani, On Some results of analysis for fuzzy metric spaces, Fuzzy Sets Syst.
  Vol.90,pp.365-368,1997.
  [5] B. Singh, S. Jain, Weak compatibility and fixed point theorems in fuzzy metric spaces, Ganita,
  56(2),pp.167-176, 2005.
  [6] M. A. Al-Thagafi and N.Shahzad,” A note on occasionally weakly compatible maps”, Int. J. Math. Anal. 3(2),pp. 55-58,2009.
  [7] H. Huang, S. Radenovi´c, and T. Doˇsenovi´c, “Some common fixed point theorems on c-distance
  in cone metric spaces over banach algebras,” Applied and Computational Mathematics, vol.14, no.
  2, pp. 180–193, 2015
  [8] T.L. Hicks, B.E. Rhoades, Fixed point theory in symmetric spaces with application to probabilistic spaces,
  Non-linear analysis, Vol.6,pp. 331-344,1999.
  [9] H. Huang, S. Hu, B. Z. Popovi´c, and S. Radenovi´c, “Common fixed point theorems for four
  mappings on cone b-metric spaces over Banach algebras,” Journal of Nonlinear Science and
  Applications, vol. 9, no. 6, pp. 3655–3671, 2016.
  [10] Dr. R Krishnakumar and Nagaral Pandit Sanatammappa, study on two Non-Archimedian Menger Probability metric space, International Journal of Statistics and Applied Mathematics 1(3),pp.01-04,2016
  [11] R. Krishnakumar, R. Livingston, “Study On Fixed Point Theorems in Complete Cone Metric Spaces” International Journal of Innovative Research in Science, Engineering and Technology, Volume 6, Issue 7,pp.13526 – 13531,2017.
  [12] R. Krishnakumar , K. Dinesh , D. Dhamodharan, Some fixed point theoremsφ- ψ weak contraction on Fuzzy Metric spaces, International Journal of Scientific Research in Mathematical and Statistical Sciences, Volume-5, Issue-3, pp.146- 152, June (2018)
  [13] O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11,pp. 326-334,1975.
  
  [14] W. A. Wilson, On semi-metric spaces, American Journal of Mathematics, 53(2),pp.361-373,1931.

  Citation
  R. Krishnakumar and Nagaral Pandit Sanatammappa, "Fixed Point Theorem of Generalized Contraction Function in Fuzzy Symmetric Spaces," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.105-110, 2019

• Open Access   Article

  Retainer of Mathematics Instead of Proficient

  V. Dahiya

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.111-114, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.111114

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  Abstract
  In all school subjects, mathematics is such that its status is unique - but its position is contrary. At one hand, it is considered an essential and compulsory part of school education from Class 1 to class 10th. On the other hand, it is one of the toughest subject in school, and because of this, the fear of failure in the students is pervasive, even among adult/parents. According to the ASER report in 2018, the percentage of children in the age group of 6 to 14 years who did not enroll in the school has dropped from three percent to 2.8 percent. The paradox described above produces many questions. Some of them are: What is math and why should we teach it to school? Does the problem arising with school mathematics relate to the nature of mathematics, or the way it is taught, or is there anything to do with it? Can all children read mathematics at a certain level? What kind of mathematics should we teach in school? And how should you teach? In this paper it may be ambitious to try to answer all the questions above.

  Key-Words / Index Term
  ASER, Mathematics, Education System

  References
  [1]. Annual Status of Education Report (ASER) 2018.
  [2]. M. Hughes, “Children and Number: Difficulties in Learning Mathematics”, B. Blackwell Publication, 1986.

  Citation
  V. Dahiya, "Retainer of Mathematics Instead of Proficient," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.111-114, 2019

• Open Access   Article

  Encryption with Key stream via Short vectors

  P. Anuradha Kameswari and S B T Sundari Katakam

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.115-120, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.115120

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  Abstract
  In this paper we describe a cryptosystem with keystream for Encryption, based on concatenation of short vectors of lattices corresponding to quadratic forms, that are good rational approximations of √z, for z a non-perfect square number. This cryptosystem is employed by both Symmetric key and Public key cryptosystem that is based on the hard mathematical problem of retrieving an irrational number from the convergents obtained as short vectors.

  Key-Words / Index Term
  Encryption, key stream, short vector, LLL algorithm, Lattice reduction, quadratic form

  References
  [1] Tom M. Apostol, Introduction to Analytical Number Theory, Springer-Verlag, New York, 1976.
  [2] J. Buchmann, Introduction to cryptography , Springer-Verlag , New York, 2004
  [3] David M. Burton, Elementary Number Theory , Sixth Edition, Tata McGraw-Hill Publishing company limited, New Delhi, India, 2008.
  [4] H.Cohen, A course in Computational Algebraic Number Theory, Graduate Texts in Math.138. Springer ,1996.
  [5] S.C. Coutinho, The Mathematics of Ciphers, University Press (India) Private Limited, 2003.
  [6] H. Davenport, The Higher Arithmetic, Eighth edition, Cambridge University Press, United Kingdom, 2008.
  [7] Jeffery Hoftstein, Jill Pipher, Joseph H. Silverman, An Introduction to Mathematical Cryptography, Second edition, Springer.
  [8] P. Anuradha Kameswari, L. Praveen Kumar, Encryption on Elliptic Curves over Z_pq with Arithmetic on E(Z_pq) via E(Z_p) and E(Z_q), IOSR Journal of Mathematics (IOSR-JM), e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 10, Issue 6 Ver. V (Nov - Dec. 2014), 21-29.
  [9] P. Anuradha Kameswari, B. Ravitheja, "Encryption Using Lucas sequences L(Δ,pq) With Arithmetic on L(Δ,pq) via L(Δ,p) and L(Δ,q), International Journal of Scientific Research in Mathematical and Statistical Sciences Volume 6, No.1, pp.178-186, 2019
  [10] A. M. Kane, "On the use of Continued Fractions for Stream Ciphers", In Proceedings of Security and Management, Las Vegas, USA , 2009.
  [11] Neal Koblitz, A course in Number Theory and cryptography, Graduate Texts in Mathematics, second edition, Springer.
  [12] A.K.Lenstra, H.W. Lenstra and L. Lovasz, Factoring Polynomials with Rational coefficients, Math.Ann.261, Springer - Verlag , pp. 515-534, 1982.
  [13] Phong Q. Nguyen, Brigitte Valléee (Eds.), The LLL Algorithm, Survey and Applications, Springer, 2010.
  [14] Nigel P.Smart, The Algorithmic Resolution of Diophantine Equations, London Mathematical Society, Student Texts 41, 1998.

  Citation
  P. Anuradha Kameswari and S B T Sundari Katakam, "Encryption with Key stream via Short vectors," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.115-120, 2019

• Open Access   Article

  K-Forcing Number of Some Graphs and Their Splitting Graphs

  Premodkmuar K P , Charles Dominic and Baby Chacko

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.3 , pp.121-127, Jun-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.121127

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  Abstract
  Amos et al. instigated the idea of k-forcing number of a graph. For a graph G=(V,E) and a subset Fof G, the vertices in F are called initially colored black vertices and the vertices in V(G)-F are called initially non colored black vertices or white vertices. Then the set F is a k-forcing set of a graph G if all vertices in G will become colored black after applying the subsequent color changing rule. Color changing rule: If a black colored vertex is adjacent to atmost k-white vertices, then the white vertices change to be colored black. The cardinality of a smallest k-forcing set is known as the k-forcing number of the graph G and is represented as Z_k (G). This work is intended to investigate the k-forcing number of the splitting graph of a graph in which k≥2.

  Key-Words / Index Term
  Zero Forcing Number, K-forcing Number and Splitting Graph

  References
  [1]. E. Sampathkumar and H.B. Walikaer, “On the splitting graph of a graph”, Journal of Karnatak University Science, Vol.25, pp.13-16, 1981.
  [2]. K. P. Premodkumar, Ch. Dominic and B. Chacko, “On the Zero Forcing Number of Graphs and Their Splitting Graphs”, Algebra and Discrete Mathematics, Accepted, 2018.
  [3]. H. Van der Holst et al., “Zero forcing sets and the minimum rank of graphs”, Linear Algebra and its Applications, Vol. 428, pp. 1628–1648, 2008.
  [4]. D. Burgarth and V. Giovannetti. “Full control by locally induced relaxation”, Physical review letters, Vol. 99, Issue.10,pp 100501-100504, 2007.
  [5]. D. Burgarth, V. Giovannetti, L. Hogben, S. Severini, and M. Young. “Logic circuits from zero forcing”.arXiv preprint arXiv:1106.4403, 2011.
  [6]. L. Hogben, D. Burgarth, D. Alessandro, S.Severini, and M. Young. “Zero forcing, linear and quantum controllability for systems evolving on networks”, IEEE Transactions on Automatic Control, vol. 58, Issue. 9, pp. 2349-2354, 2013.
  [7]. D Amos, Y caro, R Davila and R Pepper, “Upper bounds on the k-forcing number of a graph”, Discrete Applied Mathematics,Vol.181, pp 1–10, 2015.
  [8]. E. Sampathkumar, M.S. Subramanya and Ch. Dominic, “3-Consecutive vertex Coloring of a Graph”, Proc. ICDM 2008, RMS-Lecture Notes Series Vol.13, pp.161-170, 2010.
  [9]. H. Whitney, “Congruent graphs and the connectivity of graphs”. Amer. J. Math. Vol.54,pp 150-168, 1932.
  [10]. F. Harary, Graph Theory, Addison-Wesely, Massachusethes, 1969.
  [11]. R. Frucht and F. Harary, “On the Corona of two Graphs”, A equationes Math, Vol. 4 pp. 322-325, 1970.
  [12]. M. Hladnik., Marusic, D. and Pisanski, “Cyclic Haar Graphs”, Discrete. Mathematics. Vol 244, pp.137-153, 2002.
  [13]. Y Zhao, L Chen and H Li., “On Tight Bounds for the k-Forcing Number of a Graph”, Bull. Malays. Math. Sci. Sco., DOI 10.1007/s40840-017-0507-7.
  

  Citation
  Premodkmuar K P , Charles Dominic and Baby Chacko, "K-Forcing Number of Some Graphs and Their Splitting Graphs," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.3, pp.121-127, 2019