A note on the augmented Zagreb index of cacti with fixed number of vertices and cycles

Akbar Ali, Akhlaq A. Bhatti

Abstract


Recent studies show that augmented Zagreb index (AZI) possess the best correlating ability among the various well known topological indices for predicting the certain physicochemical properties ofparticular types of molecules. Hence it is meaningful to study the mathematical properties of AZI, especially bounds and characterization of the extremal elements of well known graph families. Let C_{n,k} be the family of all cacti with k cycles and n>3 vertices. In the present note, the element of the class C_{n,k} having minimum AZI is characterized. Moreover, some structural properties of the graph(s) having maximum AZI value over the collection C_{n,0} are also reported.

Keywords


Augmented Zagreb index; cactus graph; topological index.

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References


Ali, A., Bhatti, A.A. & Raza, Z. (2014). A note on the zeroth-order

general Randic? index of cacti and polyomino chains. Iranian Journal of

Mathematical Chemistry, 5:143-152.

Ali, A., Bhatti, A.A. & Raza, Z. (2016a). The augmented Zagreb

index, vertex connectivity and matching number of graphs. Bulletin of

the Iranian Mathematical Society, 42(2):417-425.

Ali, A., Raza, Z. & Bhatti, A.A. (2016b). On the augmented Zagreb

index. Kuwait Journal of Science, 43(2):48-63.

Ashrafi, A.R., Dehghan-Zadeh, T., Habibi, N. & John, P.E. (2016).

Maximum values of atom-bond connectivity index in the class of

tricyclic graphs. Journal of Applied Mathematics and Computing,

(1):511-527.

Chen, S. (2016). Cacti with the smallest, second smallest, and third

smallest Gutman index. Journal of Combinatorial Optimization,

(1):327-332.

Du, J., Su, G., Tu, J. & Gutman, I. (2015). The degree resistance

distance of cacti. Discrete Applied Mathematics, 188:16-24.

Dimitrov, D. (2016). On structural properties of trees with minimal

atom-bond connectivity indexII: Bounds on B1- and B2-branches.

Discrete Applied Mathematics, 204:90-116.

Estrada, E., Torres, L., Rodrguez, 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.

Furtula, B., Gutman, I. & Dehmer, M. (2013). On structuresensitivity

of degree-based topological indices. Applied Mathematics

and Computation, 219:8973-8978.

Furtula, B., Graovac, A. & Vuki evic?, D. (2010). Augmented Zagreb

index. Journal of Mathematical Chemistry, 48:370-380.

Gutman, I. & Furtula, B. (Eds.) (2010). Novel Molecular Structure

Descriptors - Theory and Applications. vols. I-II, Univ. Kragujevac,

Kragujevac.

Gutman, I. & Toovic?, J. (2013). Testing the quality of molecular

structure descriptors: Vertex-degree-based topological indices. Journal

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

Gutman, I., Furtula, B., Ahmadi, M.B., Hosseini, S.A., Salehi

Nowbandegani, P. & Zarrinderakht, M. (2013). The ABC index

conundrum. Filomat, 27:1075-1083.

Gutman, I., Furtula, B. & Elphick, C. (2014). Three new/old

vertex-degree-based topological indices. MATCH Communications in

Mathematical and in Computer Chemistry, 72:617-632.

Gutman, I., Furtula, B. & Ivanovic?, M. (2012). Notes on trees with

minimal atom-bond connectivity index. MATCH Communications in

Mathematical and in Computer Chemistry, 67:467-482.

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

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

connected graphs. MATCH Communications in Mathematical and in

Computer Chemistry, 67:483-494.

Lin, W., Ma, C., Chen, Q., Chen, J., Gao, T. & Cai, B. (2015). Parallel

search trees with minimal ABC index with MPI + OpenMP. MATCH

Communications in Mathematical and in Computer Chemistry, 73:337-

Lu, M., Zhang, L. & Tian, F. (2006). On the Randic? index of cacti.

MATCH Communications in Mathematical and in Computer Chemistry,

:551-556.

Palacios, J.L. (2014). A resistive upper bound for the ABC index.

MATCH Communications in Mathematical and in Computer Chemistry,

:709-713.

Raza, Z., Bhatti, A.A. & Ali, A. (2016). More on comparison between

first geometric-arithmetic index and atom-bond connectivity index.

Miskolc Mathematical Notes, 17(1):561-570.

Trinajstic?, N. (1992). Chemical Graph Theory (2nd revised ed.). CRC

Press, Florida.

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.


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