Turkish Journal of Analysis and Number Theory

ISSN (Print): 2333-1100

ISSN (Online): 2333-1232

Frequency: bimonthly

Editor-in-Chief: Mehmet Acikgoz, Feng Qi, Cenap Özel

Website: http://www.sciepub.com/journal/TJANT

Google-based Impact Factor: 2.54   Citations

Article

On the Bounds of the First Reformulated Zagreb Index

1Department of Mathematics, University of Haifa, 3498838 Haifa, Israel

2Institute for Computer Science, Friedrich Schiller University Jena, Germany

3Department of Mathematics, Velammal Engineering College, Surapet, Chennai-66, Tamil Nadu, India

4Department of Mathematics, Sacred Heart College, Tirupattur-635601, Tamil Nadu, India


Turkish Journal of Analysis and Number Theory. 2016, 4(1), 8-15
doi: 10.12691/tjant-4-1-2
Copyright © 2016 Science and Education Publishing

Cite this paper:
T. Mansour, M. A. Rostami, E. Suresh, G. B. A. Xavier. On the Bounds of the First Reformulated Zagreb Index. Turkish Journal of Analysis and Number Theory. 2016; 4(1):8-15. doi: 10.12691/tjant-4-1-2.

Correspondence to: E.  Suresh, Department of Mathematics, Velammal Engineering College, Surapet, Chennai-66, Tamil Nadu, India. Email: sureshkako@gmail.com

Abstract

The edge version of traditional first Zagreb index is known as first reformulated Zagreb index. In this paper, we analyze and compare various lower and upper bounds for the first reformulated Zagreb index and we propose new lower and upper bounds which are stronger than the existing and recent results [Appl. Math. Comp. 273 (2016) 16-20]. In addition, we prove that our bounds are superior in comparison with the other existing bounds.

Keywords

References

[1]  M. Ali Rostami, H. Martin Bucker and A. Azadi, Illustrating a Graph Coloring Algorithm Based on the Principle of Inclusion and Exclusion Using GraphTea, LNCS, Springer. 8719 (2014) 514-517.
 
[2]  A.R. Ashrafi, T. Došlić and A.Hamzeha, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010) 1571-1578.
 
[3]  M. Bianchi, A. Cornaro, J. L. Palacios, A. Torriero, New bounds of degree-based topological indices for some classes of c-cyclic graphs. Discrete App. Math. 184 (2015) 62-75.
 
[4]  G.B.A. Xavier, E. Suresh and I. Gutman, Counting relations for general Zagreb indices, Kragujevac J. Math. 38 (2014) 95-103.
 
[5]  K. C. Das, K. Xu and J. Nam, Zagreb indices of graphs, Front. Math. China 10 (2015) 567-582.
 
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[6]  N. De, Some bounds of reformulated Zagreb indices, Appl. Math. Sci. 6 (2012) 5005-5012.
 
[7]  S. Fajtlowicz S, On conjectures of graffiti II, Congr. Numer. 60 (1987), 189-197.
 
[8]  B. Fortula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) 1184-1190.
 
[9]  I. Gutman, B. Ruščić, N. Trinajstić and C. F. Wilcox, Graph theory and molecular orbitals, XII. Acyclic polyenes, J. Chem. Phys. 62 (1975) 3399-3405.
 
[10]  A. Ilić, M. Ilić and B. Liu, On the upper bounds for the first Zagreb index, Kragujevac Journal of Math. 35 (2011) 173-182.
 
[11]  A. Ilić and B. Zhou, On reformulated Zagreb indices, Discrete App. Math. 160 (2012) 204-209.
 
[12]  S. Ji, X. Li and B. Huo, On Reformulated Zagreb Indices with Respect to Acyclic, Unicyclic and Bicyclic Graphs, MATCH Commun. Math. Comput. Chem. 72 (2014) 723{732.
 
[13]  X. Li and J. Zheng, A unified approach to the extremal trees for different indices, MATCH Commun. Math. Comput. Chem. 54 (2005) 195-208.
 
[14]  T. Mansour and C. Song, The a and (a,b)- Analogs of Zagreb Indices and Coindices of Graphs, Intern. J. Combin. (2012) ID 909285.
 
[15]  A. Miličević, S. Nikolić and N. Trinajstić, On reformulated Zagreb indices, Mol. Divers. 8 (2004) 393-399.
 
[16]  E.I. Milovanović, I.Ž. Milovanović, E.Ć. Dolićanin and E. Glogić, A note on the first reformulated Zagreb index, Appl. Math. Comp. 273 (2016) 16-20.
 
[17]  G. Su, L. Xiong, L. Xu and B. Ma, On the maximum and minimum first reformulated Zagreb index with connectivity bat most k, FILMAT 25 (2011) 75-83.
 
[18]  K. Xu and K.C. Das, Some extremal graphs with respect to inverse degree, Discrete App. Math. (2015).
 
[19]  B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010) 210-218.
 
[20]  B. Zhou and N. Trinajstić, Some properties of the reformulated Zagreb indices, J. Math. Chem. 48 (2010) 714-719.
 
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Article

Schur-Convexity for a Class of Symmetric Functions

1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China


Turkish Journal of Analysis and Number Theory. 2016, 4(1), 16-19
doi: 10.12691/tjant-4-1-3
Copyright © 2016 Science and Education Publishing

Cite this paper:
Shu-Hong wang, Shu-Ping Bai. Schur-Convexity for a Class of Symmetric Functions. Turkish Journal of Analysis and Number Theory. 2016; 4(1):16-19. doi: 10.12691/tjant-4-1-3.

Correspondence to: Shu-Hong  wang, College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China. Email: shuhong7682@163.com

Abstract

In this paper, we discuss Schur convexity for a class of symmetric functions.

Keywords

References

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[2]  I. Franjić, J. Pečarić, Schur-convexity and the Simpson formula, Appl. Math. Lett., (2011).
 
[3]  N. Elezović, J. Pečarić, A note on Schur-convex functions, Rocky Mountain J. Math., 30 (3) (2000) , 853-856.
 
[4]  X. Zhang, Y. Chu, Convexity of the integral arithmetic mean of a convex function, Rocky Mountain J. Math. , 40 (3) (2010), 1061-1068.
 
[5]  A. M. Marshall, I. Olkin, B C. Arnold, Inequalities: Theory of Majorization and its Application (Second Edition). Springer New York, (2011). 101.
 
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[6]  H. N. Shi, J. Zang, Schur convexity, Schur geometric and Schur harmonic convexity of dual form of a class symmetric functions. Journal Mathematical & Inequalities, 8(2) (2014), 349-358.
 
[7]  H. N. Shi, J. Zang, Schur-convexity of dual form of some symmetric functions, Journal of Inequalities and Applications, 295 (2013), 9 papes.
 
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Article

Some Identities of Tribonacci Polynomials

1School of Studies in Mathematics, Vikram University Ujjain, (M. P.), India

2Department of Mathematical Sciences and Computer applications, Bundelkhand University, Jhansi (U. P.), India


Turkish Journal of Analysis and Number Theory. 2016, 4(1), 20-22
doi: 10.12691/tjant-4-1-4
Copyright © 2016 Science and Education Publishing

Cite this paper:
Yogesh Kumar Gupta, V. H. Badshah, Mamta Singh, Kiran Sisodiya. Some Identities of Tribonacci Polynomials. Turkish Journal of Analysis and Number Theory. 2016; 4(1):20-22. doi: 10.12691/tjant-4-1-4.

Correspondence to: Yogesh  Kumar Gupta, School of Studies in Mathematics, Vikram University Ujjain, (M. P.), India. Email: yogeshgupta.880@rediffmail.com

Abstract

The Tribonacci polynomial is famous for possessing wonderful and amazing properties. Tribonacci polynomials tn(x) defined by the recurrence relation tn+3(x)=x2tn+2(x)+xtn+1(x)+tn(x) for n0 with to(x) =o, t1(x)=1, t2(x)=x2. In this paper, we introduce some identities Tribonacci polynomials by standard techniques.

Keywords

References

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[17]  W.Goh, M.x. He, P.E. Ricci, on the universal zero attrattor of the Tribonacci- Related Polynomials, (2009).
 
[18]  Yogesh Kumar Gupta, V. H. Badshah, Mamta Singh, and Kiran Sisodiya, Generalized Additive Coupled Fibonacci Sequences of Third order and Some Identities, International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 3, March 2015, 80-85.
 
[19]  Yogesh Kumar Gupta, Kiran Sisodiya, Mamta Singh, and Generalization of Fibonacci Sequence and Related Properties, “Research Journal of Computation and Mathematics, Vol. 3, No. 2, (2015), 12-18.
 
[20]  Yogesh Kumar Gupta, V. H. Badshah, Mamta Singh, and Kiran Sisodiya, “Diagonal Function of k-Lucas Polynomials.” Turkish Journal of Analysis and Number Theory, vol. 3, no. 2 (2015): 49-52.
 
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