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Sharp bounds on certain degree based topological indices for generalized Sierpiński graphs. (English) Zbl 1434.05038
Summary: Sierpiński graphs are broadly investigated graphs of fractal nature with applications in topology, computer science and mathematics of Tower of Hanoi. The generalized Sierpiński graphs are determined by reproduction of precisely the same graph, producing self-similar graph. Graph invariant referred to as topological index is used to predict physico-chemical properties, thermodynamic properties and biological activity of chemical. In QSAR/QSPR study, these graph invariants act a key role. In this article, we studied the first, second Zagreb and forgotten indices for generalized Sierpiński graph with arbitrary base graph \(G\). Moreover, we obtained some sharp bounds with different parameters as order, size, maximum and minimum degree of \(G\) for these topological indices of generalized Sierpiński graph.
05C09 Graphical indices (Wiener index, Zagreb index, Randić index, etc.)
Full Text: DOI
[1] Akhter, S.; Imran, M., Computing the forgotten topological index of four operations on graphs, AKCE Int J Graphs Combin, 14, 70-79 (2017)
[2] An, M.; Das, K. C., First Zagreb index, k-connectivity, beta-deficiency and k-hamiltonicity of graphs, MATCH Commun Math Comput Chem, 80, 1, 141-151 (2018)
[3] Borovićanin, B.; Das, K. C.; Furtula, B.; Gutman, I., Zagreb indices: bounds and extremal graphs, MATCH Commun Math Comput Chem, 78, 1, 17-100 (2017)
[4] Che, Z.; Chen, Z., Lower and upper bounds of the forgotten topological index, MATCH Commun Math Comput Chem, 76, 635-648 (2016)
[5] Cristea, L. L.; Steinsky, B., Distances in Sierpiński graphs and on the Sierpiński gasket, Aequat Math, 85, 201-219 (2013)
[6] Das, K. C., Sharp bounds for the sum of the squares of the degrees of a graph, Kragujevac J Math, 25, 31-49 (2003)
[7] Das, K. C., On comparing Zagreb indices of graphs, MATCH Commun Math Comput Chem, 63, 433-440 (2010)
[8] Das, K. C.; Akgnes, N.; Togan, M.; Yurttas, A.; Cangl, I. N.; Cevik, A. S., On the first Zagreb index and multiplicative Zagreb coindices of graphs, Analele Stiint Univ Ovidius Constanta, 24, 1, 153-176 (2016)
[9] Das, K. C.; Gutman, I., Some properties of the second Zagreb index, MATCH Commun Math Comput Chem, 52, 103-112 (2004)
[10] Das, K. C.; Jeon, H.; Trinajstić, N., Comparison between the wiener index and the Zagreb indices and the eccentric connectivity index for trees, Discrete Appl Math, 171, 35-41 (2014)
[11] Das, K. C.; Xu, K.; Nam, J., On Zagreb indices of graphs, Front Math China, 10, 3, 567-582 (2015)
[12] Estrada-Moreno, A.; Rodríguez-Velázques, J. A., On the general Randić index of polymeric networks modelled by generalized Sierpiński graphs, Discrete Appl Math (2018)
[13] Furtula, B.; Gutman, I., A forgotten topological index, J Math Chem (2015)
[14] Gutman, I.; Jamil, M. K.; Akhter, N., Graphs with fixed number of pendent vertices and minimal first Zagreb index, Trans Comb, 4, 1, 43-48 (2015)
[15] Gutman, I.; Trinajstić, N., Graph theory and molecular orbitals. Total pi-electron energy of alternant hydrocarbons, Chem Phys Lett, 17, 535-538 (1972)
[16] Gutman, I.; Ruiščić, B.; Trinajstić, N.; Wilcox, C. F.; Phys., J. C., Graph theory and molecular orbitals, XII Acyclic Polyenes, 62, 3399-3405 (1975)
[17] Gutman, I., Degree-based topological indices, Croat Chem Acta, 86, 351-361 (2013)
[18] Horoldagva, B.; Das, K. C., Sharp lower bounds on the Zagreb indices of unicyclic graphs, Turk J Math, 39, 595-603 (2015)
[19] Hua, H.; Das, K. C., The relationship between eccentric connectivity index and Zagreb indices, Discrete Appl math, 161, 2480-2491 (2013)
[20] Horoldagva, B.; Das, K. C.; Selenge, T. A., Complete characterization of graphs for direct comparing Zagreb indices, Discrete Appl Math, 215, 146-154 (2016)
[21] Imran, M.; Hafi, S.; Gao, W.; Farahani, M. R., On topological properties of Sierpiński networks, Chaos Solitons Fractals, 98, 199-204 (2017)
[22] Javaid, I.; Benish, H.; Imran, M.; Khan, A.; Ullah, Z., On some bounds of the topological indices of generalized Sierpiński and extended Sierpiński graphs, J Inq App (2019)
[23] Klavžar, S.; Milutinovič, U.; Petr, C., 1-perfect codes in Sierpiński graphs, Bull Austral Math Soc, 66, 369-384 (2002)
[24] Milovanović, I.v.; Milovanović, M. M.; Milovanović, E. I., Remark on forgotten topological index of line graphs, Bull Inter Math Vir Inst, 7, 473-478 (2017)
[25] Moreno, A. E.; Velázquez, J. A.R., On the general Randić index of polymeric networks modelled by generalized Sierpiński graphs, Discrete Appl Math (2019)
[26] Pisanski, T.; Tucker, T. W., Growth in repeated truncations of maps, Atti Sem Mat Fis Univ Modena, 49, 167-176 (2001)
[27] Romik, D., Shortest paths in the tower of Hanoi graph and infinite automata, SIAM J Discrete Math, 20, 610-622 (2006)
[28] Teplyaev, A., Spectral analysis on infinite Sierpiński gaskets, J Funct Anal, 159, 2, 537-567 (1998)
[29] Vecchia, G. D.; Sanges, C., A recursively scalable network VLSI implementation, Future Gener Comput Syst, 4, 235-243 (1988)
[30] Yoon, Y. S.; Kim, J. K., A relationship between bounds on the sum of squares of a graph, J Appl Math Comput, 21, 233-238 (2006)
[31] Zhou, B., Zagreb indices, MATCH Commun Math Comput Chem, 52, 113-118 (2004)
[32] Zhou, B., Remarks on Zagreb indices, MATCH Commun Math Comput Chem, 57, 591-596 (2007)
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