Volume-5 , Issue-3 , Jun 2018, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  On First Hermitian-Zagreb Matrix and Hermitian-Zagreb Energy

  A. Bharali

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.136-139, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.136139

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  A mixed graph is a graph with edges and arcs, which can be considered as a combination of an undirected graph and a directed graph. In this paper we propose a Hermitian matrix for mixed graphs which is a modified version of the classical adjacency matrix related to the first Zagreb index. We also establish some results related to this newly proposed matrix. Finally we present some results on the Hermitian-Zagreb energy.

  Key-Words / Index Term
  First Zagreb Index, Hermitian Adjacency Matrix, Energy

  References
  [1] A. T. Balaban, et al., “Topological indices for structure-activity correlations”, Topics Curr. Chem., Vol. 114, pp. 21-55, 1983.
  [2] X. Chen, “On ABC eigenvalues and ABC energy”, Lin. Alg. Appl., Vol. 544, pp. 141-157, 2018.
  [3] E. Estrada, “The ABC Matrix”, J Math Chem, Vol. 55, pp. 1021-1033, 2017.
  [4] R. Gu, et al., “Skew Randic ́ matrix and Skew Randic ́ energy”, Trans. Combin., Vol. 5, No. 1, pp. 1-14, 2016.
  [5] K. Guo, B. Mohar, “Hermitian Adjacency Matrix of Digraphs and Mixed Graphs”, Journal of Graph Theory, Vol. 85, No. 1, pp. 217-248, 2017.
  [6] I. Gutman, B. Rus ̌c ̌ic ̌, N. Trinajstić, C. F. Wilcox, “Graph theory and molecular orbitals. XII. Acyclic polyenes”, J. Chem. Phys., Vol. 62, pp. 3399-3405, 1975.
  [7] I. Gutman, N. Trinajstić, “Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons”, Chem. Phys. Lett., Vol. 17, pp. 535-538, 1972.
  [8] I. Gutman, “Degree–based topological indices”, Croat. Chem. Acta, Vol. 86, pp. 351–361, 2013.
  [9] I. Gutman, “The energy of a graph”, Ber. Math. Stat. Sekt. Forschungsz. Graz. Vol. 103, pp. 1-22, 1978.
  [10] J. Liu, X. Li, “Hermitian-adjacency matrices and Hermitian energies of mixed graphs”, Lin. Alg. Appl., Vol. 446, pp. 182-207, 2015.
  [11] Y. Lu, et al., “Hermitian-Randic ́ matrix and Hermitian-Randic ́ energy of mixed graphs”, Journal of Inequalities and Applications, Vol. 54, 2017.
  [12] N. J. Rad, et al., “Zagreb Energy and Zagreb Estrada Index of Graphs”, MATCH Commun. Math. Comput. Chem. Vol. 79, pp. 371-386, 2018.
  [13] J. A. Rodri ́guez, “A spectral approach to the Randić index”, Lin. Alg. Appl., Vol. 400, pp. 339-344, 2005.
  [14] J. M. Rodri ́guez, J. M. Sigarreta, “Spectral properties of geometric-arithmetic index”, Appl. Math. Comp., Vol. 277, pp. 142-153, 2016.
  [15] J. M. Rodri ́guez, J. M. Sigarreta, “Spectral study of the Geometric-Arithmetic Index”, MATCH Commun. Math. Comput. Chem., Vol. 74, pp. 121-135, 2015.
  

  Citation
  A. Bharali, "On First Hermitian-Zagreb Matrix and Hermitian-Zagreb Energy," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.136-139, 2018

• Open Access   Article

  Approximation of the Cubic Functional Equation in Non-Archimedean Normed Spaces : Direct and Fixed Point Method

  Shalini Tomar, Nawneet Hooda

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.140-145, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.140145

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In this paper, we will consider the following form of cubic functional equation: f(kx+y) - f(x+ky) = (k-1)(k+1)2[f(x) – f(y)] – k(k-1)f(x-y) for a positive integer k greater than 2. We will investigate the Hyers-Ulam-Rassias stability of cubic functional equation using two different approaches i.e. direct and fixed point method in non-Archimedean normed spaces.

  Key-Words / Index Term
  Hyers-Ulam-Rassias stability, Cubic functional equation and non-archimedean normed spaces

  References
  [1]. S. M. Ulam,"Problems in Modern Mathematics", Science Editions, JohnWiley and Sons, New York, NY, USA,1964.
  [2]. H. Azadi Kenary,"Non-Archimedean stability of Cauchy-Jensen type functional equation ", Int. J. nonlinear analysis appl., vol. 1, issue. 2, pp. 1-10, 2010.
  [3]. J.B. Diaz ,B. Margolis,"A fixed point theorem of the alternative for contraction on a generalized complete metric space",Bull. Amer. Math. Soc.,vol. 74,pp. 305-309, 1968.
  [4]. M. Eshaghi Gordji and H. Khodaei,"Stability of Functional Equations", Lap Lambert Academic Publishing, 2010.
  [5]. M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias, and M. B. Savadkouhi,"Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces",Abstract and Applied Analysis, vol. 2009, Article ID 417473, 14 pages.
  [6]. P. Gavruta, "A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings",J. Math. Anal. Appl., vol. 184, issue. 3, pp. 431-436, 1994.
  [7]. K. Hensel, "Ubereine news Begrundung der Theorie der algebraischen Zahlen", Jahresber. Deutsch. Math., vol. 6 , pp. 83 - 88, 1897.
  [8]. N. Hooda and S. Tomar,"Approximation of the cubic functional equation in random normed spaces:direct and fixed point method",Aryabhatta Journal of Mathematics and Informatics,vol. 10,issue. 1,pp. 99-114, 2018.
  [9]. D. H. Hyers,"On the stability of the linear functional equation", Proc. Natl. Acad. Sci. USA, vol. 27, issue. 4, pp. 222-224 , 1941.
  [10]. K.W.Jun and H.M.Kim,"The generalized Hyers-Ulam-Rassias stability problem of cubic functional equation",J. Math. Anal. Appl.,vol. 27 ,pp. 867-878, 2002.
  [11]. W.A.J. Luxemburg,"On the convergence of successive approximation in the theory of ordinary differential equation",Proc. K. Ned. Aked. Wet.,Ser. A.,Indag. Math.,vol. 20,pp. 540-546, 1958.
  [12]. D. Mihet, "The stability of the additive Cauchy functional equation in non-Archimedean fuzzy normed spaces", Fuzzy Sets and Systems ,vol. 161, pp. 2206-2212, 2010.
  [13]. D. Mihet¸ and V. Radu,"On the stability of the additive Cauchy functional equation in random normed spaces",J. Math Anal Appl, vol. 343, issue. 1, pp. 567 - 572, 2008.
  [14]. A. Najati, " The generalized Hyers-Ulam-Rassias stability of a cubic functional equation", Turk. J. Math., vol. 31, issue. 4, pp. 395-408, 2007.
  [15]. J. M. Rassias,"On approximation of approximately linear mappings by linear mappings", J. Funct. Anal., vol. 46, issue. 1, pp. 126-130, 1982.
  [16]. Th. M. Rassias,"On the stability of the linear mapping in Banach spaces", Proc. Amer. Math. Soc.,vol. 72 ,pp. 297 - 300, 1978.
  [17]. M. A. Sibaha,B. Bouikhalene, and E. Elqorachi, "Ulam-Gavruta-Rassias stability of a linear functional equation", Int. J. Appl. Math. Stat., vol. 7, issue. Fe07, pp. 157-166, 2007.
  

  Citation
  Shalini Tomar, Nawneet Hooda, "Approximation of the Cubic Functional Equation in Non-Archimedean Normed Spaces : Direct and Fixed Point Method," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.140-145, 2018

• Open Access   Article

  Some Fixed Point Theorems in φ-ψ weak contraction on Fuzzy Metric Space

  R. Krishnakumar, K. Dinesh, D. Dhamodharan

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.146-152, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.146152

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In this paper, we discuss some results on fixed point theorems in φ-ψ weak contraction on fuzzy metric spaces, which are study of generalisation of some existing results are also given in the form of corollary.

  Key-Words / Index Term
  fuzzy metric space, continuous t-norm, φ-ψ weak contraction

  References
  [1] C. T. Aage, Binayak S. Choudhury and Krishnapada Das, Some fixed point results in fuzzy metric spaces using control function, Surveys in Mathematics and its Applications, 12 (2017), 23 -34.
  [2] K.Amit, and K.V Ramesh, Common fixed point theorem in fuzzy metric space using control function, Commun. Korean Math. Soc. 28 (2013), 517-526.
  [3] Arslan Hojat Ansari, R. Krishnakumar and D.Dhamodharan, Common Fixed Point of Four Mapping with Contractive Modulus on Cone Banach Space via cone C-class function, Global Journal of Pure and Applied Mathematics, 13(9) (2017), 5593–5610.
  [4] D.Dhamodharan and R. Krishnakumar, Cone S-Metric Space and Fixed Point Theorems of Contractive mappings, Annals of Pure and Applied Mathematics, 14(2)(2017), 237-243.
  [5] A.George, and P.Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), 395-399.
  [6] A.George, and P.Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90 (1997), 365-399.
  [7] M.Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27 (1988), 385-389.
  [8] V.Gregori, Samuel Morillas, and Almanzor Sapena, Examples of fuzzy metrics and applications, Fuzzy Sets and Systems, 170 (2011), 95-111.
  [9] Hsien-Chung Wu, Common Coincidence Points and Common Fixed Points in Fuzzy Semi-Metric Spaces, Mathematics, 6 (29), (2018), 1-21
  [10] Humaira, Muhammad Sarwar, and G. N. V. Kishore, Fuzzy Fixed Point Results For ϕ Contractive Mapping with
  Applications, Hindawi Complexity,2018, (2018) 12 pages,doi.org/10.1155/2018/5303815
  [11] M.S.Khan, M.Swaleh and S.Sessa, Fixed point theorems by altering distance between the points, Bull. Aust. Math. Soc.,
  30(1984), 1-9.
  [12] J.Kramosil, and J.Michalek, Fuzzy metric and Statistical metric spaces, Kybernetica, 11 (1975), 326-334.
  [13] R.Krishnakumar, K. Dinesh, Fixed Point Theorems of Fuzzy Function in Complete Metric Spaces , International Journal
  Of Innovative Research In Science, Engineering And Technology, 6(7), (2017), 13557-13562
  [14] R. Krishnakumar, K. Dinesh, Fuzzy Mapping and their Fixed Point Theorems in Complete Metric Spaces, International Journal of Scientific Research in Science, Engineering and Technology, 3(8) (2017), 241-249
  [15] B.Schweizer, and A.Sklar, Statistical metric spaces, Pacific Journal of Mathematics, 10 (1960), 313-334.
  [16] Y.Shen, Dong Qiu and Wei Chen, Fixed point theorems in fuzzy metric spaces, Applied mathematics Letters, 25 (2012),
  138-141
  [17] Krishnadhan Sarkar, Kalishankar Tiwary, “Common Fixed Point Theorems for Wealkly Compatible mapping on Cone Banach Space” IJSRMSS, Volume 5, issu-2,pp-75-79 (April 2018)
  [18] L.A.Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.
  

  Citation
  R. Krishnakumar, K. Dinesh, D. Dhamodharan, "Some Fixed Point Theorems in φ-ψ weak contraction on Fuzzy Metric Space," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.146-152, 2018

• Open Access   Article

  Minimizing the Energy Consumption Using a Secure Recursive Localization Approach for Distributed Networks in Underwater Acoustic Wireless Sensor Networks

  Vaishnavi Shukla, Awadhesh Kumar, Vijay Bhan

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.153-159, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.153159

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In wireless sensor networks, Underwater Acoustic Sensor Network (UASN) is a network of wireless sensor nodes dispensed upon an absolute area within an underwater environment, communicating along each mean via acoustic signal. It executes action as much as helping technology because a variety over underwater applications. In spite about the records centric behaviour on the UASN, on occasion the accumulated facts except its area of occurrence emerge as meaningless. Thus, localization for UASNs has turn out to be an absolutely excellent theme of research. This paper proposes an event-driven distributed localization scheme for short distance 2D UASN. The Proposed design employs a recursive localization manner of which efficiently localized nodes can work as reference node to aid within localization over the other ordinary sensor nodes. The proposed scheme employs a recursive localization procedure in which we can improve better power consumption, packet delivery ratio and delay. The simulation is done using Network Simulator-2.35.

  Key-Words / Index Term
  Anchor nodes, Acoustic Signal, Localization, UASN, Energy model, AODV, Trust Value

  References
  [1] Juan Manuel Martinez,” Planar localization of radio frequency or Acoustic sources with two parameters” presented at the 4th International Electronic Conference on Sensors and Applications, leganas, Spain.
  [2] Guangjie Han, Jinfang Jiang, Ning Sun, and Lei Shu,”Secure communication for underwater acoustic sensor networks”ISSN 0163-6804, volume 8, IEEE JOURNALS..
  [3] Miguel Ardid illium, “Acoustic Transmitters for Underwater Neutrino Telescopes In Mathematical and Statistical Sciences”, ISSN 1424-8220 mdpi journals.
  [4] Mukesh Beniwal, “Localization Techniques and Their Challengesin “ Underwater Wireless Sensor networks”, Vol. 5 (3) , 2014,4706-4710, I SSN 0975-9646 International Journal of Computer Science and Information Technologies.
  [5] Emad Felemban,” Underwater Sensor Network Applications:A Comprehensive Survey” Volume 2015, International Journal of Distributed Sensor Networks.
  [6] Pallavi G, “ A recursive approach for distributed networks in UWSN” ISSN 2395-0072, International Research Journal of Engineering and Technology.
  [7] V. Nancy Priyanka, P. Kaythry, R. Kishore."Performance analysis of Recursive Luby transform codes for error correction in underwater sensor network", 2017 International Conference on Wireless Communications, Signal Processing and Networking, (WiSPNET),2017.
  [8] Leena Pal, Pradeep Sharma, Netram Kaurav and Shivlal Mewada, "Performance Analysis of Reactive and Proactive Routing Protocols for Mobile Ad-hoc –Networks", International Journal of Scientific Research in Network Security and Communication, Vol.1, Issue.5, pp.1-4, 2013
  [9] Pedram Vahdani Amoli. "An Overview on Current Researches on Underwater Sensor Networks: Applications, Challenges and Future Trends", International Journal of Electrical and Computer Engineering (IJECE), 2016.
  [10] Preeti Sachan. "Authenticated Routing for Ad-Hoc On-Demand Distance Vector Routing Protocol", Communications in Computer And Information Science, 2011.
  [11] Han, Guangjie, Jinfang Jiang, Ning Sun, and Lei Shu. "Secure communication for underwater acoustic sensor networks", IEEE Communications Magazine, 2015.
  [12] Rajendra Aaseri ,” Trust Value Algorithm: Approach Against Packet Drop Attack In Wireless Adhoc Networks”, International Journal of Network Security & Its Applications (IJNSA), Vol.5,
  No.3, May 2013.
  [15] R. Nathiya, S.G. Santhi, "Energy Efficient Routing with Mobile Collector in Wireless Sensor Networks (WSNs)", International Journal of Computer Sciences and Engineering, Vol.2, Issue.2, pp.36-43, 2014.
  

  Citation
  Vaishnavi Shukla, Awadhesh Kumar, Vijay Bhan, "Minimizing the Energy Consumption Using a Secure Recursive Localization Approach for Distributed Networks in Underwater Acoustic Wireless Sensor Networks," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.153-159, 2018