On Generalized Atom-bond Connectivity Index of Cacti

Document Type : Research Paper


School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China


The generalized atom-bond connectivity index of a graph G is denoted by ABCa(G) and defined as the sum of weights ((d(u)+d(v)-2)/d(u)d(v))aa$ over all edges uv∊G. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, we compute sharp bounds for  ABCa index for cacti of order $n$ with fixed number of cycles and for cacti of order $n$ with given number of pendant vertices. Furthermore, we identify all the cacti that achieve the bounds.


  1. A‎. ‎Ali‎ and‎ A‎. ‎A‎. ‎Bhatti‎, ‎A note on the augmented Zagreb index of cacti with fixed number of vertices and cycles‎, ‎Kuwait J‎. ‎Sci‎. 43 (2016) 11–17‎. ‎
  2. A. R‎. ‎Ashrafi‎, ‎T‎. ‎Dehghan-Zadeh‎‎ and ‎N‎. ‎Habibi‎, ‎Extremal atom-bond connectivity index of cactus graphs‎, ‎Commun‎.Korean Math‎. ‎Soc‎. 30 (2015) 283–295‎. ‎
  3. J‎. ‎Chen‎‎ and‎ X‎. ‎Guo‎, ‎Extremal atom-bond connectivity index of graphs‎, ‎MATCH Commun. Math‎. ‎Comput‎. ‎Chem‎. 65 (2011) 713–722‎. ‎
  4. J‎. ‎Chen‎, ‎J‎. ‎Liu and‎ X‎. ‎Guo‎, ‎Some upper bounds for the atom-bond connectivity index of graphs‎, ‎Appl‎. ‎Math‎. ‎Lett‎. 25 (2012) 1077–1081‎. ‎
  5. X‎. ‎Chen and ‎G‎. ‎Hao‎, ‎Extremal graphs with respect to generalized ABC index‎, ‎Discrete Appl‎. ‎Math. 243 (2018) 115–124.
  6. K. C‎. ‎Das‎, ‎Atom-bond connectivity index of graphs‎, ‎Discrete Appl‎. ‎Math‎. 158 (2010) 1181–1188‎. ‎
  7. K. C‎. ‎Das‎, ‎I‎. ‎Gutman ‎and ‎B‎. ‎Furtula‎, ‎On atom-bond connectivity index‎, ‎Chem‎. ‎Phys‎. ‎Lett‎. 511 (2011) 452–454‎. ‎
  8. H‎. ‎Dong‎ and ‎X‎. ‎Wu‎, ‎On the atom-bond connectivity index of cacti‎, ‎Filomat 28 (2014) 1711–1717‎. ‎
  9. E‎. ‎Estrada‎, ‎Atom-bond connectivity and the energetic of branched alkanes‎, ‎Chem‎. ‎Phys‎. ‎Lett‎. 463 (2008) 422–425‎. ‎
  10. E‎. ‎Estrada‎, ‎The ABC matrix‎, ‎J‎. ‎Math‎. ‎Chem‎. 55 (2017) 1021–1033‎. ‎
  11. E‎. ‎Estrada‎, ‎L‎. ‎Torres‎, ‎L‎. ‎Rodríguez ‎and ‎I‎. ‎Gutman‎, ‎An atom-bond connectivity index‎. ‎Modelling the enthalpy of formation of alkanes‎, ‎Indian J‎. ‎Chem‎. ‎37A (1998) 849–855‎. ‎
  12. B‎. ‎Furtula‎, ‎A‎. ‎Graovac ‎and ‎D‎. ‎Vukičević‎, ‎Augmented Zagreb index‎, ‎J‎. ‎Math‎. ‎Chem‎.48 (2010) 370–380‎. ‎
  13. B‎. ‎Furtula‎, ‎A‎. ‎Graovac‎ and ‎D‎. ‎Vukičević‎, ‎Atom-bond connectivity index of trees‎, ‎Discrete Appl‎. ‎Math‎. 157 (2009) 2828–2835‎. ‎
  14. F‎. ‎Hayat and ‎B‎. ‎Zhou‎, ‎On cacti with large Mostar index‎, ‎Filomat 33 (15) (2019) ‎4865–4873‎. ‎
  15. J‎. ‎Li and ‎B‎. ‎Zhou‎, ‎Atom-bond connectivity index of unicyclic graphs with perfect matchings‎, ‎Ars Combin‎.109 (2013) 321–326‎. ‎
  16. J‎. ‎Liu‎, ‎R‎. ‎Zheng‎, ‎J‎. ‎Chen ‎and ‎B‎. ‎Liu‎, ‎The extremal general atom–bond connectivity indices of unicyclic and bicyclic graphs‎, ‎Commun‎. ‎Math‎. ‎Comput‎. ‎Chem‎. 81 (2019) 345–360‎. ‎
  17.  M‎. ‎Randić‎, ‎On characterization of molecular branching‎, ‎J‎. ‎Am‎. ‎Chem‎. ‎Soc‎. 97 (1975) 6609–6615‎. ‎
  18. R‎. ‎Xing‎, ‎B‎. ‎Zhou‎ and ‎F‎. ‎Dong‎, ‎On atom-bond connectivity index of connected graphs‎, ‎Discrete Appl‎. ‎Math‎.159 (2011) 1617–1630‎. ‎
  19. R‎. ‎Xing and ‎B‎. ‎Zhou‎, ‎Extremal trees with fixed degree sequence for atom-bond connectivity index‎, ‎Filomat 26 (2012) 683–688‎. ‎
  20. R‎. ‎Xing‎, ‎B‎. ‎Zhou ‎and ‎Z‎. ‎Du‎, ‎Further results on atom-bond connectivity index of trees‎, ‎Discrete Appl‎. ‎Math‎.‎ 158 (2010) 1536–1545‎. ‎
  21. B‎. ‎Zhou and ‎R‎. ‎Xing‎, ‎On atom-bond connectivity index‎, ‎Z‎. ‎Naturforsch‎. 66a (2011) 61–66‎. ‎
  22. B‎. ‎Zhou and ‎N‎. ‎Trinajstić‎, ‎On general sum-connectivity index‎, ‎J‎. ‎Math‎. ‎Chem‎. 47 (2010) 210–218‎. ‎
  23. B‎. ‎Zhou and ‎N‎. ‎Trinajstić‎, ‎On a novel connectivity index‎, ‎J‎. ‎Math‎. ‎Chem‎. ‎46 (2009) 1252–1270‎. ‎