# The augmented Zagreb index, vertex connectivity and matching number of graphs

Document Type : Research Paper

Authors

1 Department of Mathematics, National University of Computer and Emerging Sciences, B-Block, Faisal Town, Lahore, Pakistan.

2 Department of Mathematics, National University of Computer and Emerging Sciences, B-Block, Faisal Town, Lahore, Pakistan.and Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, United Arab Emirates.

Abstract

‎Let $\Gamma_{n,\kappa}$ be the class of all graphs with $n\geq3$ vertices and $\kappa\geq2$ vertex connectivity‎. ‎Denote by $\Upsilon_{n,\beta}$ the family of all connected graphs with $n\geq4$ vertices and matching number $\beta$ where $2\leq\beta\leq\lfloor\frac{n}{2}\rfloor$‎. ‎In the classes of graphs $\Gamma_{n,\kappa}$ and $\Upsilon_{n,\beta}$‎, ‎the elements having maximum augmented Zagreb index are determined.

Keywords

Main Subjects

#### References

A. Ali, A. A. Bhatti and Z. Raza, Further inequalities between vertex-degree-based topological indices, arXiv:1401.7511 [math.CO].
A. Ali, Z. Raza and A. A. Bhatti, On the augmented Zagreb index, arXiv:1402.3078 [math.CO].
J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976.
J. Chen, J. Liu and X. Guo, Some upper bounds for the atom-bond connectivity index of graphs, Appl. Math. Lett. 25 (2012), no. 7, 1077{1081.
K. C. Das, I. Gutman and B. Furtula, On atom-bond connectivity index, Filomat 26 (2012), no. 4, 733{738.
J. Devillers and A. T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, 1999.
D. Dimitrov, Efficient computation of trees with minimal atom-bond connectivity index, Appl. Math. Comput. 224 (2013), no. 1, 663{670.
E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008), no. 6, 422{425.
E. Estrada, L. Torres, L. Rodrهguez and I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. A 37 (1998), no. 10, 849{855.
B. Furtula, I. Gutman, M. Ivanovic and D. Vukicevic, Computer search for trees with minimal ABC index, Appl. Math. Comput. 219 (2012), no. 2, 767{772.
B. Furtula, A. Graovac and D. Vukicevic, Augmented Zagreb index, J. Math. Chem. 48 (2010), no. 2, 370{380.
I. Gutman, B. Furtula, M. B. Ahmadi, S. A. Hosseini, P. S. Nowbandegani and M. Zarrinderakht, The ABC index conundrum, Filomat 27 (2013), no. 6, 1075{1083.
I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors-Theory and Applications vols. I-II, Univ. Kragujevac, Kragujevac, 2010.
I. Gutman and J. Tosovic, Testing the quality of molecular structure descriptors: Vertex- degree-based topological indices, J. Serb. Chem. Soc. 78 (2013), no. 6, 805{810.
I. Gutman, J. Tosovic, S. Radenkovic and S. Markovic, On atom-bond connectivity index and its chemical applicability, Indian J. Chem. A 51 (2012), no. 5, 690{694.
F. Harary, Graph Theory, Addison-Wesley, Philippines, 1969.
S. A. Hosseini, M. B. Ahmadi and I. Gutman, Kragujevac trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 71 (2014), no. 1, 5{20.
Y. Huang, B. Liu and L. Gan, Augmented Zagreb index of connected graphs, MATCH Commun. Math. Comput. Chem. 67 (2012), no. 2, 483-494.
W. Karcher and J. Devillers, Practical Applications of Quantitative Structure-Activity Relationships (QSAR) in Environmental Chemistry and Toxicology, Springer, Heidelberg, 1990.
W. Lin, T. Gao, Q. Chen and X. Lin, On the minimal ABC index of connected graphs with given degree sequence, MATCH Commun. Math. Comput. Chem. 69 (2013), no. 3, 571-578.
L. Lovasz and M. D. Plummer, Matching theory [M], Ann. Discrete Math. 29, Amsterdam, 1986.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000.
R. Todeschini and V. Consonni, Molecular Descriptors for Chemoinformatics, Wiley-VCH, Weinheim, 2009.
D.Wang, Y. Huang and B. Liu, Bounds on augmented Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 209-216.

### History

• Receive Date: 22 March 2014
• Revise Date: 07 February 2015
• Accept Date: 07 February 2015