Sharp Bounds on the Augmented Zagreb Index of Graph Operations
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Authors: N. DEHGARDI AND H. ARAM
DOI: 10.46793/KgJMat2004.509D
Abstract:
Let G be a finite and simple graph with edge set E(G). The augmented Zagreb index of G is
where dG(u) denotes the degree of a vertex u in G. In this paper, we give some bounds of this index for join, corona, cartesian and composition product of graphs by general sum-connectivity index and general Randić index and compute the sharp amount of that for the regular graphs.
Keywords:
Augmented Zagreb index, general sum-connectivity index, general Randić index, graph operations.
References:
[1] A. Ali, Z. Raza and A. A. Bhatti, On the augmented Zagreb index, Kuwait J. Sci. 43 (2016), 48–63.
[2] H. Aram and N. Dehgardi, Reformulated F-index of graph operations, Commun. Comb. Optim. 2 (2017), 1–12.
[3] H. Aram, N. Dehgardi and A. Khodkar, The third ABC index of graph products, Bull. Int. Combin. Math. Appl. 78 (2016), 69–82.
[4] M. Arezoomand and B. Taeri, Zagreb indices of the generalized hierarchical product of graphs, MATCH Commun. Math. Comput. Chem. 69 (2013), 131–140.
[5] A. R. Ashrafi, T. Došlić and A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010), 1571–1578.
[6] M. Azari, Sharp lower bounds on the Narumi-Katayama index of graph operations, Appl. Math. Comput. 239 (2014), 409–421.
[7] M. Azari and A. Iranmanesh, Chemical graphs constructed from rooted product and their Zagreb indices, MATCH Commun. Math. Comput. Chem. 70 (2013), 901–919.
[8] M. Azari and A. Iranmanesh, Some inequalities for the multiplicative sum Zagreb index of graph operations, J. Math. Inequal. 9 (2015), 727–738.
[9] B. Bollobás and P. Erdós, Graphs of extremal weights, Ars Combin. 50 (1998), 225–233.
[10] J. A. Bondy and U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics 244, Springer-Verlag, London, 2008.
[11] K. C. Das, A. Yurttas, M. Togan, A. S. Cevik and I. N. Cangül, The multiplicative Zagreb indices of graph operations, J. Inequal. Appl. 90, (2013), 1–14.
[12] N. Dehgardi, A note on revised Szeged index of graph operations, Iranian J. Math. Chem. 9(1) (2018), 57–63.
[13] K. Fathalikhani, H. Faramarzi and H. Yousefi-Azari, Total eccentricity of some graph operations, Electron. Notes in Discrete Math. 45 (2014), 125–131.
[14] G. A. Fath-Tabar, B. Vaez-Zadah, A. R. Ashrafi and A. Graovac, Some inequalities for the atom-bond connectivity index of graph operations, Discrete Appl. Math. 159 (2011), 1323–1330.
[15] B. Furtula, A. Graovac and D. Vukičević, Augmented Zagreb index, J. Math. Chem. 48 (2010), 370–380.
[16] B. Furtula, I. Gutman and M. Dehmer, On structure-sensitivity of degree-based topological indices, Appl. Math. Comput. 219(1) (2013), 8973–8978.
[17] S. Hossein-Zadeh, A. Hamzeh and A. R. Ashrafi, Wiener-type invariants of some graph operations, Filomat 23 (2009), 103–113.
[18] Y. Huang, B. Liu and L. Gan, Augmented Zagreb index of connected graphs, MATCH Commun. Math. Comput. Chem. 67 (2012), 483–494.
[19] M. H. Khalifeh, H. Yusefi Azari and A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009), 804–811.
[20] K. Pattabiraman and P. Paulraja, Harary index of product graph, Discuss. Math. Graph Theory 35 (2015) 17–33.
[21] M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975), 6609–6615.
[22] G. Su and L. Xu, Topological indices of the line graph of subdivision graphs and their Schur-bounds, Appl. Math. Comput. 253 (2015), 395–401.
[23] I. Tomescu, 2-Connected graphs with minimum general sum-connectivity index, Discrete Appl. Math. 178 (2014), 135–141.
[24] D. Wang, Y. Huang and B. Liu, Bounds on augmented Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), 209–216.
[25] Z. Yarahmadi and A. R. Ashrafi, The Szeged, vertex PI, first and second Zagreb indices of corona product of graphs, Filomat 26 (2012), 467–472.
[26] Y-N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math. 135 (1994), 17–20.
[27] F. Zhan, Y. Qiao and J. Cai, Unicyclic and bicyclic graphs with minimal augmented Zagreb index, J. Inequal. Appl. 126, (2015), 1–12.
[28] B. Zhou and N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46 (2009), 1252–1270.
[29] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010), 210–218.
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