On the augmented Zagreb index

Akbar Ali, Zahid Raza, Akhlaq Ahmad Bhatti


Topological indices play an important role in Mathematical Chemistry, especially in thequantitative structure-property relationship (QSPR) and quantitative structure-activityrelationship (QSAR) studies. Recent research indicates that to predict the certainphysico-chemical properties of particular types of molecules, the augmented Zagrebindex (AZI) possess the best correlating ability among several topological indices. Themain purpose of the current study is to establish some mathematical properties of thisindex or, more precisely, to report tight bounds for the AZI of chemical bicyclic andchemical unicyclic graphs. A Nordhaus-Gaddum-type result for AZI (of connectedgraph whose complement is connected) is also derived.


Augmented Zagreb index; chemical bicyclic graph; chemical unicyclic graph; Nordhaus-Gaddum type relations; topological index.

Full Text:



Ali, A., Bhatti, A.A. & Raza, Z. (2015) Some vertex-degree-based topological indices of polyomino

chains. Journal of Computational and Theoretical Nanoscience, 12(9):2101-2107.

Ali, A., Bhatti, A.A. & Raza, Z. (2016) (in press) The augmented Zagreb index, vertex connectivity and

matching number of graphs. Bulletin of the Iranian Mathematical Society.

Ali, A. & Bhatti, A.A. (2016) (in press). A note on the augmented Zagreb index of cacti with fixed number

of vertices and cycles. Kuwait Journal of Science.

Aouchiche, M., & Hansen, P. (2013) A survey of Nordhaus-Gaddum type relations. Discrete Applied

Mathematics, 161:466-546.

Bondy, J.A. & Murty, U.S.R. (1976) Graph theory with applications. American Elsevier, New York.

Chen, J., Liu, J. & Guo, X. (2012) Some upper bounds for the atom-bond connectivity index of graphs.

Applied Mathematics Letters, 25:1077-1081.

Chen, J. & Guo, X. (2012) The atom-bond connectivity index of chemical bicyclic graphs. Applied

Mathematics-A Journal of Chinese Universities, 27:243-252.

Das, K.C., Gutman, I. & Furtula, B.( 2012) On atom-bond connectivity index. Filomat 26:733-738.

Devillers, J. & Balaban, A.T. (Eds.) (1999) Topological Indices and Related Descriptors in QSAR and

QSPR. Gordon and Breach, Amsterdam.

Dimitrov, D. (2013) Efficient computation of trees with minimal atom-bond connectivity index. Applied

Mathematics and Computation, 224:663-670.

Dimitrov, D. (2014) On structural properties of trees with minimal atom-bond connectivity index. Discrete

Applied Mathematics 172:28-44.

Estrada, E., Torres, L., Rodríguez, L. & Gutman, I. (1998) An atom-bond connectivity index: modelling

the enthalpy of formation of alkanes. Indian Journal of Chemistry-Section, A 37:849-855.

Estrada, E. (2008) Atom-bond connectivity and the energetic of branched alkanes. Chemical Physics

Letters, 463:422-425.

Furtula, B., Graovac, A. & Vukičević, D. (2009) Atom-bond connectivity index of trees. Discrete Applied

Mathematics, 157:2828-2835.

Furtula, B., Graovac, A. & Vukičević, D. (2010) Augmented Zagreb index. Journal of Mathematical

Chemistry, 48:370-380.

Furtula, B., Gutman, I., Ivanović, M. & Vukičević, D. (2012) Computer search for trees with minimal

ABC index. Applied Mathematics and Computation, 219:767-772.

Furtula, B., Gutman, I. & Dehmer, M. (2013) On structure-sensitivity of degree-based topological

indices. Applied Mathematics and Computation, 219:8973-8978.

Gutman, I. & Furtula, B. (Eds.) (2010) Novel Molecular Structure Descriptors—Theory and Applications.

vols. I-II, Univ. Kragujevac, Kragujevac, 2010.

Gutman, I. & Tošović, J. (2013) Testing the quality of molecular structure descriptors: Vertex-degreebased

topological indices. Journal of the Serbian Chemical Society, 78:805-810.

Gutman, I., Furtula, B., Ahmadi, M.B., Hosseini, S.A. & Salehi Nowbandegani, P. et al. (2013) The

ABC index conundrum. Filomat, 27:1075-1083.

Harary, F. 1969. Graph Theory. Addison-Wesley, Reading, MA.

Hosseini, S.A., Ahmadi, M.B. & Gutman, I. (2014) Kragujevac trees with minimal atom-bond

connectivity index. MATCH Communications in Mathematical and in Computer Chemistry, 71:


Huang, Y., Liu, B. & Gan, L. (2012) Augmented Zagreb index of connected graphs. MATCH

Communications in Mathematical and in Computer Chemistry, 67:483-494.

Huang, Y. & Bolian, L. (2015) Ordering graphs by the augmented Zagreb indices. Journal of Mathematical

Research with Applications, 35(2):119-129.

Imran, M., Hayat, S. & Mailk, M.Y.H. (2014) On topological indices of certain interconnection networks.

Applied Mathematics and Computation, 244:936-951.

Karcher, W. & Devillers, J. (1990) Practical Applications of Quantitative Structure-Activity Relationships

(QSAR) in Environmental Chemistry and Toxicology. Springer.

Lin, W., Gao, T., Chen, Q. & Lin, X. (2013) On the minimal ABC index of connected graphs with given

degree sequence. MATCH Communications in Mathematical and in Computer Chemistry, 69:571-

Nordhaus, E.A. & Gaddum, J.W. (1956) On complementary graphs. American Mathematical Monthly,


Todeschini, R. & Consonni, V. (2009) Molecular Descriptors for Chemoinformatics. Wiley-VCH,


Wang, D., Huang, Y. & Liu, B. (2012) Bounds on augmented Zagreb index. MATCH Communications

in Mathematical and in Computer Chemistry, 68:209-216.

Zhan, F., Qiao, Y. & Cai, J. (2015) Unicyclic and bicyclic graphs with minimal augmented Zagreb index.

Journal of Inequalities and Applications: DOI 10.1186/s13660-015-0651-2.


  • There are currently no refbacks.