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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

### Article

**On the Bounds of the First Reformulated Zagreb Index**

^{1}Department of Mathematics, University of Haifa, 3498838 Haifa, Israel

^{2}Institute for Computer Science, Friedrich Schiller University Jena, Germany

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

^{4}Department 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

*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. | ||

[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. | ||

### Article

**Schur-Convexity for a Class of Symmetric Functions**

^{1}College 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

### Keywords

### References

[1] | Y. Chu, G.Wang, X. Zhang, Schur convexity and Hadamard’s inequality, Math. Inequal. Appl., 13 (4) (2010), 725-731. | ||

[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. | ||

[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. | ||

### Article

**Some Identities of Tribonacci Polynomials**

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

^{2}Department 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

_{n}

*(x)*defined

*by the recurrence relation*

*t*

_{n+3}

*(x)=x*

^{2}

*t*

_{n+2}

*(x)+xt*

_{n+1}

*(x)+t*

_{n}

*(x) for n*≥

*0*

*with*

*t*

_{o}

*(x) =o, t*

_{1}

*(x)=1, t*

_{2}

*(x)=x*

^{2}

*.*In this paper, we introduce some identities Tribonacci polynomials by standard techniques.

### Keywords

### References

[1] | Biknell, M Hoggatt, V.E. Jr. Generalized Fibonacci, Polynomials Fibonacci Q.016, 300-303 (1978). | ||

[2] | Burrage, K generalized Fibonacci polynomials and the functional iteration of Rational Function of degree one Fibonacci Q.28, 175-180 (1990). | ||

[3] | Byrd, P.F. Expansion of Analytic Functions In polynomials associated with Fibenacci Numbers, the Fibonacci, quarterly, 1(1), (1963), 16-29. | ||

[4] | Catalon, E., Notes surla Theoric des Fractions continuous sur certain series, mem. Acad. R Belgique, 45, (1883) 1-82. | ||

[5] | Dilcher, K.A generalization of Fibonacci polynomials of integer order Fibonacci Q.16, 300-303 (1978). | ||

[6] | Djordjevi´c, G. B. and Srivastava, H. M., Some generalizations of certain sequences associated with the Fibonacci numbers, J. Indonesian Math. Soc. 12 (2006) 99-112. | ||

[7] | Djordjevi´c, G. B. and Srivastava, H. M., Some generalizations of the incomplete Fibonacci and the incomplete Lucas polynomials, Adv. Stud. Contemp. Math. 11 (2005) 11-32. | ||

[8] | He, m.x, Ricci, P.E. Asymptotic distribution of zero of weighted fibonacci polynomials, complex var. 28, 375-384 (1996). | ||

[9] | Hoggatt, V. E., Jr.: Bicknell, Marjporie (1973), “Roots of Fibonacci, Macmillan, New York (1960). | ||

[10] | Jacosthal, E, Fibonacci Polynomial and kreisteil ungsgleichugen sitzugaberichteder Berlinear, math gesells chaft, 17 (1919), 43-57. | ||

[11] | Koshy, T. Fibonacci Lucas numbers with application (willey, New York, (2001). | ||

[12] | Lucy, Joan slater “Generalized Hypergeometric functions” Cambridge University press (1966). | ||

[13] | Patel. J.M. Advanced problems and solutions, the Fibonacci quarterly, 44(1) (2006) 91. | ||

[14] | Srivastava, H. M. and. Manocha, H. L, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984. | ||

[15] | Swami, M.N.S. problem B-74, The fibonacci Quarterly, 3, (1965), 236. Solution by David Zeitlin, the Fibonacci quarterly, 4(1) (1966), 94. | ||

[16] | Vorobyou, N.N. The Fibonacci numbers D.C. Health Company, Boston (1963). | ||

[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. | ||