A note on the augmented Zagreb index of cacti with fixed number of vertices and cycles
Keywords:
Augmented Zagreb index, cactus graph, topological index.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.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.