Edge Version of Harmonic Index and Polynomial of Some Classes of Bridge Graphs
Abstract
We report explicit formulas for the edge version of harmonic indexand harmonic polynomial of several classes of bridge graphs.
References
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[2] S.H. Bertz: The bond graph, J.C.S. Chem. Commun., (1981), 818-820.
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[5] O. Favaron, M. Maheo, J. F. Sacle: Some eigenvalue properties in graphs (conjectures of Grafiti II), Discrete Math. 111 (1993), 197-220.
[6] I. Gutman: Edge versions of topological indices: I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors - Theory and Applications II , Univ. Kragujevac, Kraguje-
vac, 3, 2010.
[7] I. Gutman, E. Estrada: Topological indices based on the line graph of the molecular graph, J. Chem. Inf. Comput. Sci., 36 (1996), 541-543.
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[9] I. Gutman, Z. Tomovic: On the application of line graphs in quantitative structure-property studies, J. Serb. Chem. Soc., 65 (8) (2000), 577-580.
[10] I. Gutman, Z. Tomovic: Modeling boiling points of cycloalkanes by means of iterated line graph sequences, J. Chem. Inf. Comput. Sci., 41 (2001), 1041-1045.
[11] F. Harary, R.Z. Norman: Some properties of line digraphs, Rendiconti del Circolo Matematico di Palermo, 9 (2) (1960), 161-169.
[12] H. Hosoya: "Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons", Bulletin of the Chemical Society
of Japan, 44 (9) (1971), 2332-2339.
[13] A. Iranmanesh, I. Gutman, O. Khormali, A. Mahmiani: The edge versions of the Wiener index, MATCH Comm. Math. Comput. Chem., 61 (2009), 663-672.
[14] M.A. Iranmanesh, M. Saheli: On the harmonic index and harmonic polynomial of Caterpillars with diameter four, Iranian J. of Math. Chem., 6(1) (2015), 41-49.
[15] J. Krausz: Dmonstration nouvelle d'un thorme de Whitney sur les rseaux, Mat. Fiz. Lapok 50: (1943), 75-85.
[16] X. Li, Y. Shi: A survey on the Randic index, MATCH Commun. Math. Comput. Chem., 59(1) (2008), 127-156.
[17] X. Li, Y. Shi, L. Wang: An updated survey on the Randic index, in: B.F. I. Gutman (Ed.), Recent Results in the Theory of Randic Index, University of Kragujevac and Faculty
of Science Kragujevac, 2008, 9-47.
[18] R. Nazir, S. Sardar, S. Zafar, Z. Zahid: Edge version of harmonic polynomial and harmonic index, Journal of Applied Mathematics and Informatics, 34 (2016), 479-486.
[19] M. F. Nadeem, S. Zafar, Z. Zahid: Certain topological indicies of the line graph of subdivsion graphs, Appl. Math. Comput., 271 (2015), 790-794.
[20] M. F. Nadeem, S. Zafar, Z. Zahid: On Topological properties of the line graph of subdivision graphs of certain nanostructures, Appl. Math. Comput., 273 (2016), 125-130.
[21] M. F. Nadeem, S. Zafar, Z. Zahid: On the edge version of geometric-arithmetic index of nanocones, Studia Ubb Chemia, 61(1), (2016), 273-282.
[22] M. F. Nadeem, S. Zafar, Z. Zahid: Some Topological Indices of L(S(CNCk[n])), Punjab University Journal of Mathematics, 49 (1)(2017), 13-17.
[23] D. Plavsic, S. Nikolic, N. Trinajstic, Z. Mihalic: On the Harary Index for the Characterization of Chemical Graphs, 12 (1993), 235-250.
[24] J. Rada, R. Cruz: Vertex-degree-based topological indices over graphs, MATCH Commun. Math. Comput. Chem., 72 (2014), 603-616.
[25] T. Doslic, M.S Litz: Matchings and Independent Sets in Polyphenylene Chains, MATCH Commun. Math. Comput. Chem., 67 (2012), 313-330.
[26] R.J. Trudeau: Introduction to Graph Theory, New York: Dover Pub. (1993), 19.
[27] H. Whitney: Congruent graphs and the connectivity of graphs : American Journal of Mathematics, 54 (1) (1932), 150-168 .
[28] G. Yu, L. Feng: On connective eccentricity index of graphs, MATCH Commun. Math. Comput. Chem., 69 (2013), 611-628.
[29] L. Zhong: The harmonic index for graphs, Appl. Math. Lett., 25 (2012), 561-566.
[30] L. Zhong: The harmonic index on unicyclic graphs, Ars Combin., 104 (2012), 261-269.
[31] L. Zhong, K. Xu: The harmonic index for bicyclic graphs, Utilitas Math., 90 (2013), 23-32.
[2] S.H. Bertz: The bond graph, J.C.S. Chem. Commun., (1981), 818-820.
[3] Devillers, J. and Balaban: Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, (1999), 31.
[4] S. Fajtlowicz: On conjectures of Grafiti II, Congr. Numer. 60 (1987), 187-197.
[5] O. Favaron, M. Maheo, J. F. Sacle: Some eigenvalue properties in graphs (conjectures of Grafiti II), Discrete Math. 111 (1993), 197-220.
[6] I. Gutman: Edge versions of topological indices: I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors - Theory and Applications II , Univ. Kragujevac, Kraguje-
vac, 3, 2010.
[7] I. Gutman, E. Estrada: Topological indices based on the line graph of the molecular graph, J. Chem. Inf. Comput. Sci., 36 (1996), 541-543.
[8] I. Gutman, L. Popovic, B. K. Mishra, M. Kaunar, E. Estrada, N. Guevara: Application of line graphs in physical chemistry. Predicting surface tension of alkanes, J. Serb. Chem. Soc., 62 (1997), 1025-1029.
[9] I. Gutman, Z. Tomovic: On the application of line graphs in quantitative structure-property studies, J. Serb. Chem. Soc., 65 (8) (2000), 577-580.
[10] I. Gutman, Z. Tomovic: Modeling boiling points of cycloalkanes by means of iterated line graph sequences, J. Chem. Inf. Comput. Sci., 41 (2001), 1041-1045.
[11] F. Harary, R.Z. Norman: Some properties of line digraphs, Rendiconti del Circolo Matematico di Palermo, 9 (2) (1960), 161-169.
[12] H. Hosoya: "Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons", Bulletin of the Chemical Society
of Japan, 44 (9) (1971), 2332-2339.
[13] A. Iranmanesh, I. Gutman, O. Khormali, A. Mahmiani: The edge versions of the Wiener index, MATCH Comm. Math. Comput. Chem., 61 (2009), 663-672.
[14] M.A. Iranmanesh, M. Saheli: On the harmonic index and harmonic polynomial of Caterpillars with diameter four, Iranian J. of Math. Chem., 6(1) (2015), 41-49.
[15] J. Krausz: Dmonstration nouvelle d'un thorme de Whitney sur les rseaux, Mat. Fiz. Lapok 50: (1943), 75-85.
[16] X. Li, Y. Shi: A survey on the Randic index, MATCH Commun. Math. Comput. Chem., 59(1) (2008), 127-156.
[17] X. Li, Y. Shi, L. Wang: An updated survey on the Randic index, in: B.F. I. Gutman (Ed.), Recent Results in the Theory of Randic Index, University of Kragujevac and Faculty
of Science Kragujevac, 2008, 9-47.
[18] R. Nazir, S. Sardar, S. Zafar, Z. Zahid: Edge version of harmonic polynomial and harmonic index, Journal of Applied Mathematics and Informatics, 34 (2016), 479-486.
[19] M. F. Nadeem, S. Zafar, Z. Zahid: Certain topological indicies of the line graph of subdivsion graphs, Appl. Math. Comput., 271 (2015), 790-794.
[20] M. F. Nadeem, S. Zafar, Z. Zahid: On Topological properties of the line graph of subdivision graphs of certain nanostructures, Appl. Math. Comput., 273 (2016), 125-130.
[21] M. F. Nadeem, S. Zafar, Z. Zahid: On the edge version of geometric-arithmetic index of nanocones, Studia Ubb Chemia, 61(1), (2016), 273-282.
[22] M. F. Nadeem, S. Zafar, Z. Zahid: Some Topological Indices of L(S(CNCk[n])), Punjab University Journal of Mathematics, 49 (1)(2017), 13-17.
[23] D. Plavsic, S. Nikolic, N. Trinajstic, Z. Mihalic: On the Harary Index for the Characterization of Chemical Graphs, 12 (1993), 235-250.
[24] J. Rada, R. Cruz: Vertex-degree-based topological indices over graphs, MATCH Commun. Math. Comput. Chem., 72 (2014), 603-616.
[25] T. Doslic, M.S Litz: Matchings and Independent Sets in Polyphenylene Chains, MATCH Commun. Math. Comput. Chem., 67 (2012), 313-330.
[26] R.J. Trudeau: Introduction to Graph Theory, New York: Dover Pub. (1993), 19.
[27] H. Whitney: Congruent graphs and the connectivity of graphs : American Journal of Mathematics, 54 (1) (1932), 150-168 .
[28] G. Yu, L. Feng: On connective eccentricity index of graphs, MATCH Commun. Math. Comput. Chem., 69 (2013), 611-628.
[29] L. Zhong: The harmonic index for graphs, Appl. Math. Lett., 25 (2012), 561-566.
[30] L. Zhong: The harmonic index on unicyclic graphs, Ars Combin., 104 (2012), 261-269.
[31] L. Zhong, K. Xu: The harmonic index for bicyclic graphs, Utilitas Math., 90 (2013), 23-32.
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2017-02-14
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